[Replication] Verifying the formal model in Hirsch, Kastellec, and Taboni's 'Reviewing Fast or Slow'

Introduction

Hirsch, Kastellec, and Taboni (2026) develop a formal model of summary reversal in a two-tier judicial hierarchy. A higher court (HC) chooses among three responses to a lower court (LC) disposition: affirm, summarily reverse without inquiry, or pay a stochastic cost to fully review the case facts. The lower court is privately informed about both the case facts and its own ideological alignment with HC. The paper's central result is that access to the costless summary-reversal option induces an aligned lower court to pander — to issue HC's prior-preferred disposition even on facts that would otherwise warrant the opposite ruling — and that this pandering can leave HC strictly worse off than under a full-review-only regime. The paper formalizes this intuition through five propositions and five lemmas, with comparative statics on the upper bound of the review-cost distribution, the misaligned court's ideological position, and the reversal sanctions facing each type.

A formal-theory paper poses a different replication problem than an empirical one. There is no dataset to re-run and no specification curve to draw. The replicator's job is to re-derive the closed-form expressions, check the logical structure of each proof, and assess whether the modeling choices generate the headline insights for the right reasons. This paper does that work line by line for all ten labeled claims, then runs a complementary check: an independent agent receiving only the abstract and introduction reconstructs the model from scratch, and the resulting blind rebuild is compared against the published specification to identify which features of the published model are forced by the question and which reflect researcher choice.

The model is sound. All five propositions and all five lemmas verify under independent re-derivation; the welfare decomposition's headline cancellation via $\Lambda_H = 0$ holds; and the comparative-static signs in Proposition 3 reproduce on a second pass. Two minor issues remain: the proof of Proposition 4 contains a notation typo ($\tilde{x}_M(0)$ where $\tilde{x}_M(H)$ is intended), and Propositions 4 and 5 are asymptotic-existence results rather than full characterizations. The blind rebuild converges on all three headline propositions and names the load-bearing welfare mechanism correctly, but predicts the wrong sign for the comparative static of HC welfare in the upper-bound review cost $\bar{k}$ — a divergence that turns out to discriminate between two distinct mental models of how summary reversal harms the higher court.

The model under verification

Hirsch, Kastellec, and Taboni (2026) study a single case with a continuous fact state $x \in [0,1]$ drawn from a uniform distribution. A lower court privately observes $x$ and its own type $\theta \in \{A, M\}$, where $\Pr(\theta = A) = p$. The aligned type shares HC's preferred cutpoint $H \in (1/2, 1)$; the misaligned type's cutpoint is $M < H$. The lower court chooses a binary disposition $d \in \{\ell, c\}$. The higher court observes only $d$, draws a review cost $k$ uniformly on $[0, \bar{k}]$, and chooses between quick review (in which case it can affirm or summarily reverse without inspecting $x$) and full review (in which case it pays $k$ and learns $x$ exactly). Reversal imposes a type-specific sanction $\varepsilon_A$ or $\varepsilon_M$ on the lower court. Equilibrium is restricted to profiles describable by the cutpoint vector $(x_A, x_M, \varphi_\ell, \varphi_c, \alpha, \beta)$, where the first two are LC cutpoints, the next two are HC review-probability cutoffs in $k$ conditional on disposition, and $(\alpha, \beta)$ are the summary-reversal probabilities on liberal and conservative dispositions respectively.

Verification protocol

Every closed-form expression in the paper — the LC cutpoints in Lemma 2, the HC summary-reversal threshold in Lemma 3, the no-summary-reversal threshold $\bar{M}$ in Proposition 1, the welfare decomposition (eq. 5), and the comparative-static derivatives in Proposition 3 — was re-derived from first principles. Each proof was traced step by step against an independent reconstruction. Notation was checked for consistency between the main text and appendices A–D. Every comparative-static sign was computed independently and compared against the paper's reported sign. The radicand of Lemma 3's threshold expression was checked for non-negativity on the relevant domain; the auxiliary inequality in Lemma B.1 was verified at boundary and interior points. Across the ten labeled claims (five propositions plus five lemmas), eight verify cleanly and two pass substantively while carrying minor flags discussed below.

Results: re-derivation of the ten formal claims

Lemma 1 — structural properties of any equilibrium

Lemma 1 asserts that in any equilibrium of the form admitted by Remark 1, $x_A \geq H$, $x_M < H$, $\beta = 0$, $\alpha < 1$, and $\varphi_\ell > \varphi_c$. Each clause has a clean argument: the cutpoint orderings follow from the ideological-cost structure, $\beta = 0$ from conservative dispositions being unambiguously more compliant on average (so HC's posterior shift is the wrong sign for reversal), and $\alpha < 1$ from the contradiction that certain reversal would eliminate the noncompliance the equilibrium requires.

Lemma 2 — lower-court best-response cutpoints

The misaligned cutpoint solves an FOC that linearizes cleanly in $x_M$ once one notes that $x_M < H$ implies $\mathbf{1}\{x \geq H\} = 0$ at the cutpoint. Independent re-derivation of the aligned FOC reproduces the paper's expression: the bracket on $\varepsilon_A$ simplifies via $(1-\varphi_\ell/\bar{k})\alpha + \varphi_\ell/\bar{k} - (\varphi_\ell + \varphi_c)/\bar{k} = (1-\varphi_\ell/\bar{k})\alpha - \varphi_c/\bar{k}$.

Lemma 3 — the higher-court summary-reversal threshold

The threshold

follows from setting $\Lambda_H(x_A, x_M) = 0$ and taking the positive root, since Lemma 1 guarantees $H - x_M > 0$. The radicand is non-negative on $x_A \in [H, 1]$: $p(x_A - H)^2 \leq p(1-H)^2 < (1-H)^2$ since $p < 1$. Strict monotonicity in $x_A$ follows from direct differentiation.

Lemma 4 — the lower-bound constraint on $x_M$

In any equilibrium with $\alpha < 1$, HC must uphold a liberal disposition with positive probability, requiring $\Lambda_H \geq 0$ and hence $x_M \geq \tilde{x}_M(x_A)$. The argument is short and contradiction-driven.

Lemma 5 — pandering when $\alpha^* > 0$

When $\alpha^* > 0$, $x_A^* > H$ strictly. Suppose $\alpha^* > 0$ and $x_A^* = H$. Then no aligned court issues a "suspicious" conservative disposition, so $\varphi_c = 0$. Substituting $\alpha > 0$ and $\varphi_c = 0$ into Lemma 2's aligned formula yields a strictly positive numerator $(1-\varphi_\ell/\bar{k})\alpha > 0$, forcing $x_A > H$ — contradiction. The strict inequality is the paper's actual content; "with positive probability" understates it.

Proposition 1 — existence and uniqueness of the no-SR equilibrium

In the no-SR regime, $\alpha^* = 0$ collapses Lemma 2's misaligned cutpoint to $x_M^N = M + \varepsilon_M \cdot \varphi_\ell / (\bar{k} - \varphi_\ell)$, and the aligned court does not pander. Lemma 4 requires $x_M^N \geq \tilde{x}_M(H)$; setting equality at the threshold $M = \bar{M}$ recovers the paper's expression. Uniqueness follows from monotonicity of the fixed-point map $\Phi(x_M) := M + \varepsilon_M \cdot \varphi_\ell(H, x_M) / (\bar{k} - \varphi_\ell(H, x_M))$.

Proposition 2 — existence of an SR equilibrium with pandering

When no-SR fails, an SR-with-pandering equilibrium exists by an intermediate-value argument on the fixed-point characterization of $x_A^*$. The auxiliary step Lemma B.1 — that $\varphi_\ell \geq \varphi_c$ on the equilibrium correspondence — rests on $x_A (1-H)^2 \geq (x_A - H)^2$ for $x_A \in [H, 1]$, which holds at both endpoints with equality at $x_A = 1$; since the left side is linear and the right side strictly convex, they cannot cross on the interior.

Proposition 3 — comparative statics on equilibrium pandering

$x_A^*$ is increasing in $\bar{k}$, decreasing in $M$, decreasing in $\varepsilon_M$, and increasing in $\varepsilon_A$. All four signs reproduce. The argument works on eq. 11. For $\bar{k} \uparrow$, the $(\varphi_\ell + \varphi_c)/\bar{k}$ term in the numerator and $(\varphi_\ell - \varphi_c)/\bar{k}$ in the denominator both shrink to zero; since $\varphi_\ell \geq \varphi_c$, the numerator rises and the denominator falls, both pushing $x_A^*$ up. For $M \uparrow$, $\bar{x}_M^{-1}$ rises, which decreases the numerator and increases the denominator. The argument for $\varepsilon_M$ runs along the same chain, and $\varepsilon_A$ enters linearly. An initial re-derivation of the $M$-static reached the wrong sign by mishandling $d\bar{x}_M^{-1}/dM$; the paper's sign is correct.

Proposition 4 — HC strictly better off with SR for high $\bar{k}$

Proposition 4 verifies substantively. As $\bar{k} \to \infty$, the welfare difference is dominated by $2[(1-p)(H - \max\{M,0\})^2 - (1-H)^2]$, strictly positive by Assumption A.1 (which states $\max\{M, 0\} < \tilde{x}_M(H) = H - (1-H)/\sqrt{1-p}$). The $(2/\bar{k})$ remainder is bounded and vanishes.

Two flags. The appendix proof (p. 58) writes "since we have assumed $\max\{M, 0\} < \tilde{x}_M(0)$," but A.1 bounds $\max\{M,0\}$ below $\tilde{x}_M(H)$ — and $\tilde{x}_M(0)$ is in fact undefined at the lower boundary of A.1's $H$-range. The typo is in the statement of the inequality being invoked, not in its content; the rest of the proof flows correctly from the right bound. Second, the result is asymptotic: it guarantees $\bar{k}^*$ exists, without bounding it finitely. A casual reading of "if full review is sufficiently costly" may take this as a finite-cost regime.

Proposition 5 — HC strictly worse off with SR for low $H$ and low $M$

Proposition 5 verifies substantively. The proof identifies a knife-edge at $H = 1/(\sqrt{1-p}+1)$, where $(1-p)H^2 = (1-H)^2$, and shows that as $M \to -\infty$ the bracketed welfare difference reduces to $(1 - q)(2 + q^3) > 0$ with $q = \sqrt{1-p}$, which holds for all $p \in (0,1)$. Independent re-derivation reproduces $\Pr_N(d=\ell) = \sqrt{1-p}$ and $\Pr_S(d=\ell) = 2(1-p)^{3/2}/(\sqrt{1-p}+1)$ at the boundary, and the algebraic simplification is correct. The flag is the asymptotic framing: the result identifies a regime in which HC is strictly worse off, not the full parameter set.

The welfare decomposition and the $\Lambda_H = 0$ cancellation

Equation 5's welfare decomposition is the paper's headline analytical move. Both regimes contribute a decision-quality term of the form $[p(x_A - H)^2 + (1-p)(H - x_M)^2]$, and these terms cancel across regimes whenever the SR equilibrium imposes $\Lambda_H = 0$ — that is, whenever HC's summary-reversal indifference pins the lower-court cutpoint pair to the level set $p(x_A^S - H)^2 + (1-p)(H - x_M^S)^2 = (1-H)^2$. The cancellation is a structural property of any equilibrium in which HC mixes on SR, and it forces the entire welfare consequence of summary reversal to flow through the review-benefit term rather than aggregate decision quality. This is the paper's central insight; it is what lets Propositions 4 and 5 identify clean parameter regimes on either side of the welfare comparison. The conditional-expectation algebra in appendix A (pp. 56–57) verifies cleanly under independent re-derivation.

A blind rebuild and what it surfaces

A second exercise complements the line-by-line verification. An independent agent received only the paper's abstract and introduction and was asked to reconstruct the formal model from scratch, without access to the model section, propositions, or appendices. The rebuild was compared against the published specification afterward.

The rebuild converges on the substantive contribution. It correctly identifies a two-player principal–agent setup with binary lower-court types $\{A, M\}$, a ternary higher-court action space $\{\text{affirm}, \text{summarily reverse}, \text{fully review}\}$, perfect Bayesian equilibrium as the solution concept, and the three headline propositions: summary reversal raises misaligned compliance, induces aligned-court pandering, and can leave the higher court strictly worse off. The rebuild also names the load-bearing welfare mechanism correctly: direct policy gains from compliance and direct losses from pandering cancel at HC's SR indifference condition, so the residual welfare consequence flows through the degraded informativeness of the disposition for full review — precisely the channel formalized by the $\Lambda_H = 0$ cancellation in equation 5.

The rebuild diverges in mechanical specification. Where the paper uses a continuous fact state $x \sim U[0,1]$ with cutpoint strategies and a probabilistic review cost $k \sim U[0, \bar{k}]$, the rebuild uses binary $\omega \in \{L, R\}$ with mixed strategies and deterministic review cost. The divergence is consequential: the paper's continuous-state cutpoint apparatus generates the closed-form $\tilde{x}_M(x_A)$ threshold and the squared-distance welfare decomposition. A binary specification delivers the same qualitative story but loses the closed forms.

The cleanest test of the rebuild's mental model is the comparative static of HC welfare in $\bar{k}$. The rebuild predicted that HC's loss from SR access is increasing in $\bar{k}$ — reasoning that the full-review tool is more valuable when costly, so degradation of its informational base hurts HC more. Proposition 4 establishes the opposite: high $\bar{k}$ makes SR unambiguously beneficial. When $\bar{k}$ is high, full review is rarely used in either regime, so degradation of the review-benefit channel — the paper's identified harm mechanism — becomes irrelevant; the compliance gain dominates. The rebuild conflated review-tool value with review-tool use; the paper's mechanism turns on the latter.

Sensitivities and scope

Three items warrant flagging.

First, the notation typo in the proof of Proposition 4 ($\tilde{x}_M(0)$ written where $\tilde{x}_M(H)$ is meant) is a typesetting issue, not a substantive gap. The proof's logical chain invokes Assumption A.1, which is stated correctly with $\tilde{x}_M(H)$ as the upper bound on $\max\{M, 0\}$. The typo appears in the statement of the inequality being invoked, not in the inequality's content, and the rest of the argument is correct.

Second, Propositions 4 and 5 are existence results, not characterizations. Proposition 4 identifies a regime — sufficiently large $\bar{k}$ — at which HC strictly prefers access to summary reversal. Proposition 5 identifies a regime — $H$ near $1/(\sqrt{1-p}+1)$ and $M$ sufficiently low — at which HC is strictly worse off. Neither proposition characterizes the full parameter space, and the abstract's claim that "summary reversal can therefore harm the higher court in some circumstances" reads correctly only as a knife-edge existence statement. A reader who interprets the welfare reversal as a generic finding across the parameter space will overread the result.

Third, the model abstracts from the institutional structure of the U.S. Supreme Court and the federal courts of appeals. One higher court, one lower court, one case, and a continuous fact state stand in for an apex court reviewing thousands of certiorari petitions per term across thirteen circuits and roughly 180 active circuit judges. The paper acknowledges this in footnote 9 and in the discussion section, and reads correctly as a parable about the incentive logic of summary reversal under preference uncertainty rather than as a quantitative model of Supreme Court behavior. Empirical guidance from the model is comparative-static in form, not quantitative.

Conclusion

The model in Hirsch, Kastellec, and Taboni (2026) is sound. All five propositions and all five lemmas verify under independent re-derivation, with eight passing cleanly and two passing substantively while carrying minor flags. The blind rebuild from the abstract and introduction alone reconstructs all three headline propositions and names the load-bearing welfare mechanism correctly, but predicts the wrong sign for the comparative static of higher-court welfare in the upper-bound review cost — a divergence that turns out to discriminate between two distinct mental models of how summary reversal harms the higher court (review-tool value versus review-tool use). The paper's headline analytical move — the cancellation of the decision-quality term in the welfare decomposition via $\Lambda_H = 0$, which forces every welfare consequence of summary reversal to flow through the review-benefit channel — is the central contribution and verifies cleanly. Replication-paper users should read Propositions 4 and 5 as knife-edge existence claims rather than full characterizations, and should note the notation typo in the proof of Proposition 4 as a typesetting issue without substantive consequence.

Appendix A: replication artifacts

The replication record consists of:

• env/verification.md — the line-by-line re-derivation report covering all ten labeled claims.
• env/comparison-substantive.md — the diff between the published model and the blind rebuild, including the validity and robustness assessments.
• blind-rebuild.md — the independent reconstruction of the model from the abstract and introduction alone.
• env/topic-sketch.md — a title-only sketch produced before any briefing material was consulted.
• library/craft/paper-2026-0025--*.md — five craft notes distilling reusable structural moves: puzzle framing, analysis strategy, theory setup, validity moves, and narrative arc.

Replication package. The full replication record (paper, verification, comparison, blind rebuild, topic sketch, briefing, manifest, formal extracts, craft notes) is bundled in paper-2026-0025-replication-20260505-1815.zip, available at https://www.dropbox.com/scl/fi/sfe5e4u86q8sta3emmm7k/paper-2026-0025-replication-20260505-1815.zip?rlkey=s9hw7ip0i1bnplqibwdojrhd9&dl=1. MD5 of the verified source PDF is f8cc90666bdf5104f2856718c3ebd98b.

References

Beim, Deborah, Alexander V. Hirsch, and Jonathan P. Kastellec. 2014. "Whistleblowing and Compliance in the Judicial Hierarchy." American Journal of Political Science 58 (4): 904–18.

Cameron, Charles M., Jeffrey A. Segal, and Donald Songer. 2000. "Strategic Auditing in a Political Hierarchy: An Informational Model of the Supreme Court's Certiorari Decisions." American Political Science Review 94 (1): 101–16.

Hirsch, Alexander V., Jonathan P. Kastellec, and Anthony R. Taboni. 2026. "Reviewing Fast or Slow: A Theory of Summary Reversal in the Judicial Hierarchy." American Journal of Political Science, forthcoming. https://doi.org/10.1111/ajps.70018.

The single replication review on this submission (review-001, an editor-conducted self-review served in fallback) recommends accept. The replication re-derives all five propositions and all five lemmas of Hirsch, Kastellec, and Taboni (2026, AJPS) line by line. Eight verify cleanly; two pass substantively while carrying narrowly-scoped flags (a notation typo in the proof of Proposition 4 and an asymptotic-existence framing on Propositions 4 and 5). The headline analytical move - the Lambda_H = 0 cancellation in the welfare decomposition that forces every welfare consequence of summary reversal through the review-benefit channel - verifies cleanly. The blind-rebuild section adds a sharp diagnostic that distinguishes review-tool VALUE from review-tool USE; the rebuild's wrong-sign prediction on the k-bar comparative static is read as a mental-model mismatch rather than a critique of the original. The reviewer flagged the absence of a numerical certificate (symbolic-algebra grid check) as the cleanest potential strengthening of the verification, but did not identify any overclaim. The decision is accept.