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Information and voting: Evidence from Peru's 2026 presidential election
\title{\Large\bfseries Information and voting:\\[6pt]
Evidence from Peru's 2026 presidential election}
\author{
Marcelo Gallardo\textsuperscript{*}
\qquad
Nicolas Velarde\textsuperscript{\dag}
\qquad
{Cristina Gutarra\textsuperscript{\ddag}}
}
\date{\today}
\maketitle
\thispagestyle{empty}
\footnotetext[1]{Department of Mathematics, Pontificia Universidad Cat\'olica del Per\'u (PUCP); Department of Economics, University of California, Berkeley. Contact: [email removed].}
\footnotetext[2]{Department of Economics, Pontificia Universidad Cat\'olica del Per\'u (PUCP); Centro de Investigaci\'on de la Universidad del Pac\'ifico (CIUP). Contact: [email removed].}
\footnotetext[3]{Pontificia Universidad Cat\'olica del Per\'u (PUCP). Contact: [email removed].}
\setcounter{footnote}{0}
\begin{abstract}
\noindent We study how election-night flash estimates shape voting in
Peru's fragmented 2026 presidential election. We exploit a natural
experiment: on 12~April~2026, 187 polling tables across 13 voting
centers failed to install, and the \emph{Jurado Nacional de Elecciones}
(JNE) extended voting for the affected $\approx\!55 000$ electors to
Monday 13~April. These voters cast ballots after observing the Ipsos and
Datum flash estimates; otherwise comparable Sunday voters did not. A
Bayesian-updating model of multi-candidate plurality voting frames the
analysis, yielding predictions about vote reallocation toward the three
candidates the estimates rendered viable---L\'opez Aliaga, S\'anchez,
and Nieto. We estimate treatment effects on candidate vote shares at both the \emph{acta} level and the acta-weighted polling-station level, comparing treated and control \emph{locales de votaci\'on} matched on pre-treatment covariates. How flash estimates reshape voting is of first-order
importance for Peru, given its institutional instability and high
political volatility over the past decade.
\end{abstract}
\noindent\textbf{JEL Classifications}: D72, D82, D83, C21, C93
\noindent\textbf{Keywords}: information design, Bayesian updating,
strategic voting, exit polls, natural experiment, cardinality matching.
\noindent \textbf{AI disclosure}: We used Anthropic's Claude (Opus 4.7 and 4.8) to assist with prose editing, code verification, and LaTeX formatting, and Refine (refine.ink) to generate referee-style feedback on the manuscript. All substantive research decisions, analysis, and conclusions are the authors' own.
\noindent\small\textit{Preliminary draft. All errors are our own. This paper is strictly theoretical and empirical, and does not reflect the political views or preferences of the authors.}
\newpage
\setcounter{page}{1}
\section{Introduction}\label{sec:intro}
In multi-candidate plurality elections with weak party identification, late-arriving public signals about candidate viability can change voters' minds. Peru's 2026 presidential first round provides a near-ideal setting in which to identify these effects. The election was extraordinarily fragmented, with thirty-six candidates on the ballot. On election night, the two leading polling firms (Ipsos and Datum Internacional) released their \emph{flash electoral} estimates within minutes of one another, agreeing on Keiko Fujimori as the leading candidate but \emph{disagreeing} on the identity of the second-place finisher. Datum named Rafael L\'opez Aliaga; Ipsos named Roberto S\'anchez. A third candidate, Jorge Nieto, was within the statistical margin of error in both estimates.
A logistical failure delivered the natural experiment. On 12~April, 187~polling tables in 13~voting centers could not be installed because the contractor failed to deliver electoral materials on time. The \emph{Jurado Nacional de Elecciones} (JNE) extended voting for the affected $\approx\!55 000$ electors to Monday 13~April. These \emph{tomorrow} voters cast their ballots after observing the flash estimates and the partial \emph{actas} count released through the night; otherwise comparable Sunday voters did not.
This paper has three components. First, we present the political and institutional context that makes Peruvian voters unusually responsive to viability signals: a decade of executive--legislative collapse, a near-complete absence of stable party brands, and a documented voter logic of rejection rather than affiliation (the \emph{mal menor}). Second, we develop a Bayesian-updating model of voting in which two pollsters issue noisy public signals about the identity of the runoff entrant. The model delivers a tractable formulation in which each voter's choice probability is a multinomial logit in which the posterior viability of a candidate is interacted with the voter's expressive match to him. Because the instrumental pull of viability is scaled by expressive affinity, the flash reallocates votes toward candidates who are both viable and ideologically acceptable to the local electorate---so a nationally viable but expressively distant candidate gains where his support is concentrated and stays flat elsewhere. The candidates whose viability the flash estimates raise are L\'opez Aliaga, S\'anchez, and Nieto.
Third, the empirical section exploits the JNE ruling to identify the
treated \emph{locales de votaci\'on}, builds a new socioeconomic
dataset from INEI Redatam census data and 2021 results, and
estimates the average treatment effect on the treated by cardinality
matching \citep{ZubizarretaParedesRosenbaum2014} and a within-group regression estimator.
We find that the informational shock raised the vote share of the
candidates that the flash estimates rendered viable for the
runoff among Monday voters
relative to comparable Sunday locales, consistent with the
viability-driven strategic reallocation the model predicts.
Our central contribution is empirical: exploiting the JNE ruling as a
natural experiment, we identify the causal effect of overnight flash
estimates on Monday voters through cardinality matching at the
polling-place level (Peru's \emph{local de votaci\'on}) and a
within-group regression estimator, using a new dataset that
characterizes these polling places socioeconomically from block-level
(\emph{manzana}) INEI Redatam census data merged with ONPE
\emph{actas} for 2021 and 2026. Adapting the single- and
competing-sender frameworks of \citet{Kamenica2011} and
\citet{GentzkowKamenica2017} to a multi-candidate plurality election
with two public signals that may disagree, the information-design model
casts the treatment as a \citet{Blackwell1953} improvement in the
public experiment and thereby gives the matching estimator its
interpretation as the behavioral value of that improvement
(Section~\ref{subsec:bridge-id}).
Our framework abstracts from information channels beyond the two flash estimates, such as rapid counts, television, and social-media commentary on platforms like X or Instagram, which plausibly exert their own influence on voter beliefs. This is a deliberate scope restriction rather than a claim of irrelevance; extending the analysis to these channels is left for future work.
Section~\ref{sec:context} sets out the political, institutional, and literature context; Section~\ref{sec:evidence} documents the 2026 empirical environment and the natural experiment; Section~\ref{sec:model} develops the information-design model; Section~\ref{sec:hypotheses} states the predictions; Section~\ref{sec:method} the empirical strategy; and Sections~\ref{sec:results}--\ref{sec:conclusion} report results and conclude.
\section{Background and literature}\label{sec:context}
\subsection{Peruvian political crisis (2016--present)}
To understand the depth of the crisis, it is necessary to trace its
roots beyond 2016. Following the collapse of the Fujimori regime in
November 2000, Alberto Fujimori's daughter Keiko Fujimori rebuilt the
Fujimorist movement under the banner of \emph{Fuerza Popular}
(Popular Force). Already under Ollanta Humala (2011--2016),
Fujimorist congressional blocs developed a pattern of
constraining executive action, establishing legislative supremacy
as a de facto norm \citep{Forbes2017}. This was facilitated by the
loosely worded ``permanent moral incapacity'' clause of
Article~113 of the 1993 Constitution, which lets Congress remove
the president without a threshold equivalent to high crimes,
rendering the legislature more powerful than the executive and
turning vacancy motions into weapons of ordinary political
competition \citep{Eguiguren2017}.
The 2016 elections crystallized these tensions. Pedro Pablo
Kuczynski (PPK) won the runoff by fewer than fifty thousand votes
over Keiko Fujimori, while Popular Force secured an absolute
congressional majority---a centre-right executive facing a
Fujimorist legislature with the numbers and constitutional tools to
block, censure, and remove the president \citep{Mauro2026}.
Entangled in the Odebrecht scandal, PPK resigned in March 2018
ahead of an imminent impeachment vote. His successor Vizcarra
dissolved Congress in 2019 after a second rejected confidence vote,
triggering snap elections, but was himself removed in November 2020
by a vacancy vote condemned by left-leaning sectors
\citep{BarrenecheaVergara2023}. Vizcarra was later sentenced to
fourteen years for accepting bribes as governor of Moquegua
\citep{Reuters2025Vizcarra} and implicated in the
\emph{Vacunagate} scandal, having received COVID-19 vaccines
outside the authorized trial protocol \citep{Lancet2021Vacunagate}.
Protests over Vizcarra's removal forced his replacement Manuel
Merino to resign after five days; Congress then appointed Francisco
Sagasti to lead a transitional government to the 2021 elections, in
which Pedro Castillo narrowly defeated Keiko Fujimori. Castillo's
presidency was acutely fragile---nearly eighty ministerial changes
in seventeen months, persistent corruption allegations, relentless
obstruction---and, facing a third vacancy vote in December 2022, he
attempted an unconstitutional self-coup and was removed and
arrested within hours \citep{BarrenecheaVergara2023}.
Vice-president Dina Boluarte assumed the presidency and aligned with
the conservative bloc, triggering months of violent Andean protests
that left more than sixty dead. Under her government congressional
factions consolidated control over key institutions, including the
Constitutional Tribunal and Public Ministry, and democratic-quality
indices downgraded Peru to hybrid-regime status \citep{EIU2025}; a
worsening security crisis and low approval led Congress to remove
her in October 2025.
Her successor Jos\'e Jer\'i lasted four months before being censured
over a corruption scandal involving undisclosed meetings with
Chinese businessmen. Congress then elected Jos\'e Mar\'ia Balc\'azar
as its president, who assumed the presidency by constitutional
succession. His appointment drew mutual accusations between
\emph{Renovaci\'on Popular} and \emph{Fuerza Popular}, each blaming
the other for enabling the ascent of a congressman from the
left-wing \emph{Per\'u Libre} \citep{RPP2026Balcazar}, and coincided
with a decline in vote intention for Rafael L\'opez Aliaga, leader of
\emph{Renovaci\'on Popular} \citep{Infobae2026LopezAliagaCae}. This
brought to nine the number of individuals to have held the office
since 2016.
Therefore, as Peru approached the 2026 elections, it did so amid prolonged political
instability. Human Rights Watch reports that measures passed by Congress
in 2024 weakened judicial independence and constrained corruption and
organized-crime investigations \citep{HRW2026Peru}, in a context where
most recent presidents and many legislators faced criminal allegations
\citep{BarrenecheaVergara2023}.
\subsection{Political parties and the electorate in Peru}\label{sec:parties}
Peru represents one of Latin America's most pronounced cases of
party system collapse. No major party has recently consolidated a stable and major
national presence, developed genuine ideological coherence, or
sustained meaningful linkages with citizens across electoral cycles. What exists instead is a succession of electoral vehicles, each
organized around a candidate rather than a program, each dissolving
or reconstituting itself after every election
\citep{MainwaringSu2021, BarrenecheaVergara2023}.
This institutional weakness is neither recent nor accidental. Its
origins trace to the 1992 \emph{autogolpe}, which dismantled a party
system that had been gradually consolidating throughout the 1980s
\citep{Tanaka2006}. The 1993 Constitution, designed to reduce the
role of parties as political intermediaries, produced a paradoxical
outcome: not a closed system that excluded citizens, but an
excessively open one that left no stable organizations between the
population and power. The result was what \cite{Tanaka2006} describes as the
``omnibus'' phenomenon, in which national parties operate from a
small nucleus in Lima with no organic territorial presence,
recruiting local operators before each election in purely
transactional arrangements that dissolve once voting ends
\citep{Tanaka2006}. Peru has consequently recorded among the highest
levels of electoral volatility in Latin America
\citep{MainwaringSu2021}, with its leading parties changing almost
entirely between electoral cycles.
The representational consequences are substantial. Each electoral cycle
effectively begins from zero, with little accumulation of institutional
experience and no correction of past failures, and parties cannot be
held accountable for their commitments because many cease to exist
before the next election. This fragility has deep historical roots:
since the founding of the republic, political life has been concentrated
in a Lima-based elite, and parties never developed genuine national
structures or deep societal roots \citep{Chavez2022}.
In the absence of stable partisan attachments, Peruvian voters have
reorganized their electoral behavior around a different logic:
rejection rather than affiliation \citep{Mainwaring2006}. Citizens
deploy their ballots not toward the candidate they support, but
against the one they most fear reaching power. This phenomenon has a consistent empirical record. The negative
partisanships of \emph{anti-aprismo} and \emph{anti-fujimorismo}
remained stable across the 2011, 2016, and 2021 elections, even as
the positive vehicles carrying those votes collapsed one after
another \citep{Melendez2019}. Humala, Kuczynski, and their
successors each attracted votes defined not by full loyalty but by the
imperative of blocking a feared adversary, a logic \cite{Melendez2019} terms the \emph{mal menor} (lesser evil). Experimental research on
political cue-taking in Peru confirms the underlying mechanism:
outgroup cues consistently decrease support for policy positions,
while ingroup cues produce no corresponding increase, revealing a
political psychology organized more around opposition than
affiliation \citep{delaCerda2025}.
This is the structural context that makes strategic vote-switching
not merely possible but rational. Where positive identification with
any candidate is weak and provisional, a voter's first-round choice
remains permanently vulnerable to revision the moment a different
vote would more effectively block the feared adversary from
advancing to the second round.
\subsection{Electoral framework}
Peru's electoral system is governed by Ley N°~26859 and administered
by three constitutionally autonomous institutions. The
\emph{Jurado Nacional de Elecciones} (JNE) functions as the supreme
electoral tribunal, responsible for overseeing legality, resolving
disputes, and proclaiming official results; its decisions in
electoral matters are final and unappealable. The \emph{Oficina
Nacional de Procesos Electorales} (ONPE) bears operational
responsibility for organizing and executing elections: it deploys
decentralized offices throughout the country, appoints
polling-station coordinators through public competition, and
coordinates with the police and armed forces to guarantee order on
election day. A third body, the \emph{Registro Nacional de
Identificaci\'on y Estado Civil} (RENIEC), maintains the voter
registry and issues the \emph{Documento Nacional de Identidad} (DNI),
without which no citizen may cast a vote.
General elections are held every five years on the second Sunday of
April (first round), and \emph{voting is mandatory} for citizens
between the ages of 18 and 70 (optional thereafter); failure to vote
is sanctioned by a monetary fine and entails temporary administrative
restrictions until the obligation is regularized. Presidential
candidates must win an outright majority of valid votes; if none
does, a runoff is held within thirty days between the two leading
candidates. Congressional elections take place simultaneously. The
2026 election marked a structurally significant change: for the
first time since 1990, voters elected a bicameral Congress
comprising 130 deputies and 60 senators. Deputies, as the lower
chamber, are tasked with presenting legislation and overseeing the
Cabinet, while senators, as the upper chamber, review bills, approve
key institutional appointments, and hold the power to remove the
president. Access to seats in either chamber is conditioned on an electoral threshold (\emph{valla electoral}) that a party must clear on two counts simultaneously: it must obtain at least five percent of the valid votes cast nationwide for that chamber, and it must be allotted, in the seat-distribution procedure, a number of seats equal to at least five percent of the chamber's legal membership---seven of the 130 deputies or three of the 60 senators. This dual requirement is designed to limit fragmentation in a historically atomized party system, though it carries the trade-off of potentially reducing effective representation when a significant share of the electorate votes for parties that fail to clear the threshold.
Voting is organized at the level of the individual \emph{mesa de
sufragio} (polling table), where ballots are cast, counted, and
recorded in official \emph{actas electorales} transmitted to ONPE's
central tally. The polling table is therefore the foundational unit
of the electoral chain, and any disruption at this level---such as
a failure to install a table on time---directly interrupts the
exercise of suffrage.
\subsection{Related literature}\label{subsec:literature}
Our paper draws on four literatures. The first is information design and
Bayesian updating \citep{Kamenica2011, Bergemann2019}, which studies
how the signals available to a decision-maker can be structured; nearer
our setting are \citet{Alonso2016}, \citet{Chan2019}, and---closest, with
Datum and Ipsos issuing potentially conflicting calls---
\citet{GentzkowKamenica2017} on competing senders. We borrow its
apparatus (signal structure, Bayesian updating, and the
\citet{Blackwell1953} comparison of experiments) but not its premise of a
strategic sender: we neither model pollsters as designing estimates to
sway voters nor take any stance on their intent. What the framework
supplies instead are the definitions and structure for our object of
interest---the natural experiment assigns Sunday and Monday voters to two
experiments over the identity of the runoff entrant, and we estimate the
behavioral value of the Blackwell improvement from the coarser
(pre-flash) to the finer (post-flash) experiment
(Section~\ref{subsec:bridge-id}).
To take this structure to behavior, we turn to a second literature on
choice modeling. The random-utility multinomial logit \citep{Train2009}
microfounds the model, mapping the expressive and viability components
into individual choice probabilities. Because the ballot is secret,
however, we never observe individual choices: our outcomes are aggregate
returns at the \emph{acta} level. The logit therefore plays a structural
role, disciplining the aggregate vote shares we observe rather than being
fit to individual data.
The methodological core of the paper belongs to a third literature, on
matching for causal inference. We use cardinality matching on
pre-treatment covariates \citep{ZubizarretaParedesRosenbaum2014,
ViscontiZubizarreta2018}, which recovers the ATT on treated locations
under selection on observables, estimated with within-group
difference-in-means and regression estimators \citep{PageLenardKeele2020}.
Finally, two empirical literatures provide background rather than method.
The first studies media and political updating \citep{DellaVigna2007,
Enikolopov2011, Gentzkow2014}; its closest antecedent to our design is
\citet{Morton2015}, who exploits a French natural experiment to identify
exit-poll-induced turnout and bandwagon effects via the geographic and
temporal staggering of exposure to a public signal. The second concerns
voting in weakly institutionalised democracies, where \citet{Lupu2016},
\citet{Dargent2015}, \citet{Vergara2018}, and \citet{Levitsky2022} ground
the strategic voting under the \emph{mal menor} logic of
Section~\ref{sec:parties}.
\section{The 2026 election and the flash-estimate shock}\label{sec:evidence}
\subsection{Candidates and pre-electoral polls}\label{subsec:positioning}
The 2026 ballot featured 36 candidates spanning the full ideological
range (Table~\ref{tab:ideology}; \citealp{TuVotoPeru2026}), and the
field was exceptionally fragmented. Fujimori (\emph{Fuerza Popular}) led
on a free-market, \emph{mano dura}, socially conservative platform, but
no other candidate pulled clear: a pack of contenders competed closely
for the second runoff slot, spanning the spectrum. On the right,
L\'opez Aliaga (\emph{Renovaci\'on Popular}) carried the most explicitly
hard-right discourse on security, religion, and social values; on the
left, S\'anchez (\emph{Juntos por el Per\'u}) ran a consistently
left-wing platform---protectionist, statist, and progressive; and
Belmont, \'Alvarez, and Nieto held intermediate or off-axis positions,
Nieto in particular running as a ``fresh option'' outside the
traditional left--right divide, with L\'opez Chau also among the leading
group. Given this ideological spread but weak programmatic
differentiation among the front-runners, voter choice is more plausibly
driven by perceived viability than by policy, consistent with the
\emph{mal menor} logic of Section~\ref{sec:parties}.
In the polls, three features defined the pre-electoral environment.
Vote intention was historically low (Figure \ref{fig:simulacros_bar}): Fujimori ran between roughly $14\%$
and $19\%$ in the final month, while the candidates competing for second
place clustered in a tight 7--13\% band, with five or six within a few
points of one another. The race was volatile, with rankings shifting
substantially between waves---Ipsos and Datum ran the two longest
regular series (Figures~\ref{fig:ipsos_reg} and~\ref{fig:datum_reg}).
And the pollsters disagreed on who occupied second place, a disagreement
that persisted into their election-night flash estimates
(Section~\ref{subsec:flash}). The six official JNE debates appear to
have moved intentions, with \'Alvarez and Belmont gaining and L\'opez
Aliaga declining \citep{ipsos2026preelect}.\footnote{Series compiled
from the \emph{La Encerrona} tracker,
\url{https://laencerrona.pe/2026/02/10/encuestas}; Peruvian law
prohibits poll publication in the week before the election.}
\subsection{The flash estimates and the Monday vote}\label{subsec:flash}
The first round produced no clear front-runner: the four leading
candidates finished within six points (Fujimori $17.06\%$, S\'anchez
$12.04\%$, L\'opez Aliaga $11.90\%$, Nieto $11.03\%$), in a
geographically fragmented result---an electoral ``archipelago''
splitting Lima and the north coast between the two right-of-centre
candidates, the Andes toward S\'anchez, and the urban south toward
Nieto \citep{Martinelli2026archipielago}. With barely a point
separating the second runoff slot, its occupant was unresolved on
election night.
At 18:00 on 12~April, Ipsos and Datum released their \emph{flash
electoral} estimates (Table~\ref{tab:flashpolls}). Both placed
Fujimori first but disagreed on second: Datum put L\'opez Aliaga
second ($12.8\%$); Ipsos put S\'anchez second ($12.1\%$) with Belmont
third ($11.8\%$). The trajectory from the three Ipsos \emph{simulacros}
to the flash showed S\'anchez and Belmont surging and \'Alvarez
declining (Figure~\ref{fig:simulacros_bar}). The estimates thus
confirmed Fujimori's passage while leaving the second slot open---a potential right-versus-right runoff, but with S\'anchez registering as
a viable left-of-center contender in Ipsos and the centrist Nieto
within the margin of error in both estimates, so that the candidates
in contention for the single remaining slot spanned the full
spectrum: L\'opez Aliaga on the radical right, Nieto at the center,
and S\'anchez on the left.
\section{The model}
\label{sec:model}
This section develops the framework that organizes the empirical
analysis. Its premise is that the Ipsos and Datum flash estimates are
public signals about the identity of the runoff entrant, and that such
a signal can in principle shift a voter's choice by revising her beliefs
about which candidates are viable. We microfound this channel at the
level of the individual voter. The objective is operational: a
closed-form expression for the probability that voter~$i$ chooses
candidate~$c$ after observing the two flash estimates, whose
within-table aggregate is the effect our matching estimator identifies.
We first model each flash estimate as a noisy public signal and derive
the voter's posterior over the runoff entrant; we then embed that
posterior in a random-utility model of individual choice in which the
instrumental value of a viable candidate is scaled by the voter's
expressive match to him.
\subsection{Primitives and notation}\label{subsec:primitives}
The state space $\Theta$ indexes the identity of the second-place
finisher. Fujimori's lead in first place is uncontested across all
major pre-electoral polls and in both flash estimates, so uncertainty
is confined to who advances with her to the runoff. Guided by the
flash data of Section~\ref{subsec:flash}---in which Datum names
L\'opez Aliaga, Ipsos names S\'anchez, and Nieto polls within a
percentage point of the named candidates in both estimates---we
restrict attention to the three candidates the flash estimates render
viable:
\begin{equation}\label{eq:theta-set}
\Theta \;\triangleq\; \{\text{Roberto S\'anchez},\;
\text{Rafael L\'opez Aliaga},\;
\text{Jorge Nieto}\} \;\triangleq\; \{\text{RS},\; \text{RLA},\; \text{JN}\}.
\end{equation}
A belief is a probability distribution on $\Theta$; the set
of beliefs is the unit simplex
$$
\Delta(\Theta) = \left\{\mu \in \mathbb{R}_+^{K} : \sum_{\theta} \mu(\theta) = 1\right\},
$$
so that $\mu_j \in \Delta(\Theta)$ for every voter $j$. The
prior $\mu_0 \in \Delta(\Theta)$ is the belief held before any
signal is observed.
\subsection{Bayesian updating with one signal}\label{subsec:bayes-one}
A signal---or experiment\footnote{An experiment in the sense
of \citet{Blackwell1953} is a pair consisting of a finite signal
space and a family of probability distributions over it, one for
each state of the world.}---is a pair $(S, \pi)$ consisting of a
finite signal space~$S$ and a Markov kernel $\pi : \Theta \to \Delta(S)$.
\begin{definition}[Posterior belief]\label{def:posterior}
Upon observing $s \in S$ with $\mathbb{P}_{\mu_0}(s) > 0$, the posterior
is
\begin{equation}\label{eq:bayes}
\mu(\theta \mid s) \;=\; \frac{\pi(s \mid \theta)\,\mu_0(\theta)}{\sum_{\theta' \in \Theta} \pi(s \mid \theta')\,\mu_0(\theta')}.
\end{equation}
\end{definition}
We model each pollster's flash estimate as a symmetric
\emph{$\eta$-noisy} experiment: $S = \Theta$ and
\begin{equation}\label{eq:eta-noisy}
\pi^\eta(s \mid \theta) \;=\;
\begin{cases}
1 - \eta & \text{if } s = \theta,\\[2pt]
\dfrac{\eta}{K - 1} & \text{if } s \neq \theta,
\end{cases}
\qquad \eta \in [0, 1 - 1/K).
\end{equation}
Nature draws $\theta$; the experiment announces it correctly with
probability $1 - \eta$ or, with probability $\eta$, announces one
of the $K-1$ wrong states uniformly. The single number $\eta$
absorbs both sampling and non-sampling error in a way that admits a
clean Blackwell ordering: $\pi^{\eta_1} \succeq_B \pi^{\eta_2}$
whenever $\eta_1 \le \eta_2$ \citep[in the sense of][]{Blackwell1953}.
\begin{lemma}[Posterior under one $\eta$-noisy signal]\label{lem:single-post}
Fix $\mu_0 \in \Delta(\Theta)$, $\eta \in [0, 1 - 1/K)$, and
$s^* \in \Theta$ the realised signal value. Then
\begin{equation}\label{eq:single-post}
\mu(\theta \mid s^*) \;=\;
\begin{cases}
\dfrac{(1-\eta)\,\mu_0(s^*)}{Z(\mu_0, s^*, \eta)} & \theta = s^*,\\[12pt]
\dfrac{\eta\,\mu_0(\theta)/(K-1)}{Z(\mu_0, s^*, \eta)} & \theta \neq s^*,
\end{cases}
\end{equation}
with normalising constant
$Z(\mu_0, s^*, \eta) = (1-\eta)\,\mu_0(s^*) + \tfrac{\eta}{K-1}\!\left(1 - \mu_0(s^*)\right)$.
\end{lemma}
\begin{proof}
Bayes' rule \eqref{eq:bayes}, applied with prior $\mu_0$ and
likelihood $\pi^\eta$, gives
\[
\mu(\theta \mid s^*)
\;=\;
\frac{\pi^\eta(s^* \mid \theta)\,\mu_0(\theta)}
{\sum_{\theta' \in \Theta} \pi^\eta(s^* \mid \theta')\,\mu_0(\theta')}
\]
for every $\theta \in \Theta$. Splitting the denominator according
to whether $\theta'$ equals $s^*$, and using
$\pi^\eta(s^* \mid s^*) = 1 - \eta$ together with
$\pi^\eta(s^* \mid \theta') = \eta/(K-1)$ for $\theta' \neq s^*$ from
\eqref{eq:eta-noisy},
\[
\sum_{\theta' \in \Theta} \pi^\eta(s^* \mid \theta')\,\mu_0(\theta')
\;=\;
(1-\eta)\,\mu_0(s^*) \;+\; \frac{\eta}{K-1} \sum_{\theta' \neq s^*} \mu_0(\theta').
\]
Since $\mu_0 \in \Delta(\Theta)$,
$\sum_{\theta' \neq s^*} \mu_0(\theta') = 1 - \mu_0(s^*)$, so the
denominator equals
$Z(\mu_0, s^*, \eta) = (1-\eta)\,\mu_0(s^*) + \tfrac{\eta}{K-1}\bigl(1 - \mu_0(s^*)\bigr)$,
which is strictly positive on $\eta \in [0,\,1-1/K)$ because both
summands are non-negative and at least one is strictly positive.
Substituting $\pi^\eta(s^* \mid \theta)$ from \eqref{eq:eta-noisy}
into the numerator yields the two branches of
\eqref{eq:single-post}: $(1-\eta)\,\mu_0(s^*)/Z$ when $\theta = s^*$,
and $\bigl(\eta/(K-1)\bigr)\,\mu_0(\theta)/Z$ when $\theta \neq s^*$.
Non-negativity of each branch is immediate, and summing over
$\theta \in \Theta$ recovers $Z/Z = 1$, confirming
$\mu(\,\cdot \mid s^*) \in \Delta(\Theta)$.
\end{proof}
\begin{remark}[Relative informativeness of the two flash estimates]\label{rem:eta-calib}
We treat $\eta_f$ as a model primitive. Ipsos has the stronger record on
prior second-place calls, so we set $\eta_I < \eta_D$, taking it as the
sharper experiment; by~\eqref{eq:eta-noisy} this gives the Blackwell
ranking $\pi^{\eta_I} \succeq_B \pi^{\eta_D}$. The 2026 result---S\'anchez
did reach the runoff, as Ipsos had it---corroborates the ordering ex
post, but this was not yet known when the \emph{ma\~nana} voters saw the
flash, so it plays no role in their beliefs.
\end{remark}
\subsection{Two signals}\label{subsec:two-signal}
The 2026 flash produced calls $(s_D^*, s_I^*) = (\text{RLA},
\text{RS})$. We assume the two signals are conditionally independent
given the state.
\begin{assumption}[Conditional independence]\label{ass:cond-indep}
Conditional on $\theta$,
$$
\mathbb{P}(s_D, s_I \mid \theta) = \pi^{\eta_D}(s_D \mid \theta)\,\pi^{\eta_I}(s_I \mid \theta).
$$
\end{assumption}
The assumption is supported by the operational independence of the
two firms' samples and weighting procedures; we discuss it further
in Section~\ref{sec:method}.
\begin{proposition}[Joint posterior]\label{prop:joint-post}
Under Assumption~\ref{ass:cond-indep},
\begin{equation}\label{eq:joint-post}
\mu(\theta \mid s_D^*, s_I^*)
\;=\; \frac{\pi^{\eta_D}(s_D^* \mid \theta)\,\pi^{\eta_I}(s_I^* \mid \theta)\,\mu_0(\theta)}
{\sum_{\theta' \in \Theta} \pi^{\eta_D}(s_D^* \mid \theta')\,\pi^{\eta_I}(s_I^* \mid \theta')\,\mu_0(\theta')}.
\end{equation}
\end{proposition}
\begin{proof}
View $(s_D, s_I)$ as a single signal taking values in the product
space $S \times S$, with joint kernel
$\pi(s_D, s_I \mid \theta) \coloneqq \mathbb{P}(S_D = s_D, S_I = s_I \mid \theta)$.
Bayes' rule, applied to this joint signal gives
\begin{equation}\label{eq:bayes-joint}
\mu(\theta \mid s_D^*, s_I^*)
\;=\; \frac{\pi(s_D^*, s_I^* \mid \theta)\,\mu_0(\theta)}
{\sum_{\theta' \in \Theta} \pi(s_D^*, s_I^* \mid \theta')\,\mu_0(\theta')}.
\end{equation}
Assumption~\ref{ass:cond-indep} states that, conditional on
$\theta$, the two flash signals are independent, so the joint
kernel factorizes as
\[
\pi(s_D, s_I \mid \theta)
\;=\;
\pi^{\eta_D}(s_D \mid \theta)\,\pi^{\eta_I}(s_I \mid \theta)
\]
for every $\theta \in \Theta$ and every $(s_D, s_I) \in S \times S$.
Substituting this factorisation into both the numerator and the
denominator of \eqref{eq:bayes-joint} yields \eqref{eq:joint-post}.
Strict positivity of the denominator follows from
$\eta_D, \eta_I \in [0,\,1-1/K)$, which guarantees
$\pi^{\eta_D}(s_D^* \mid \theta'), \pi^{\eta_I}(s_I^* \mid \theta') > 0$
for every $\theta' \in \Theta$, so that the full-support prior
condition is preserved in the posterior.
\end{proof}
\begin{example}
[Numerical illustration with $(\eta_D, \eta_I) = (0.40, 0.30)$ and uniform prior.]
With $K = 3$ and $\mu_0 \equiv 1/3$, the joint likelihoods evaluated
at $(s_D^*, s_I^*) = (\text{RLA},\text{RS})$ are
\[
\begin{aligned}
L(\text{RLA}) &= (1-\eta_D)\cdot(\eta_I/2) = 0.60 \times 0.15 = 0.090,\\
L(\text{RS}) &= (\eta_D/2)\cdot(1-\eta_I) = 0.20 \times 0.70 = 0.140,\\
L(\text{JN}) &= (\eta_D/2)\cdot(\eta_I/2) = 0.20 \times 0.15 = 0.030,
\end{aligned}
\]
with sum $0.260$. Equation~\eqref{eq:joint-post} therefore yields
\[
\mu\!\left(\cdot \mid s_D^* = \text{RLA},\, s_I^* = \text{RS}\right)
\;\approx\; \big(\,\underbrace{0.346}_{\text{RLA}},\;
\underbrace{0.538}_{\text{RS}},\;
\underbrace{0.115}_{\text{JN}}\,\big).
\]
The sharper Ipsos call still leads, but less decisively now that the two
experiments are closer in precision: posterior mass on S\'anchez is about
$0.54$, while L\'opez Aliaga retains about $0.35$ as Datum's call, and
Nieto---named by neither pollster but observed close to both---retains
about $0.12$. All three remain substantially above zero, so the discordant flash only
redistributes viability mass within $\Theta$ rather than concentrating it on a
single named candidate: since $\mu_0\in\Delta(\Theta)$ already assigns
$\{\mathrm{RS},\mathrm{RLA},\mathrm{JN}\}$ collective probability one, every
posterior holds that total at one and moves mass only among the three. The long
tail's non-viability is not a result of the update but of the \emph{ex ante}
restriction of the state space to $\Theta$
justified by the polling and flash data; this is what
sets $q_c(\mu)=0$ for every $c\notin\Theta$, at the prior and at every posterior
alike.
\end{example}
\subsection{From utility to vote probabilities}\label{subsec:utility}
Candidates are indexed by $c \in \mathcal{C}$ (the full 36-candidate
ballot of Section~\ref{subsec:positioning}, of which $\Theta$ is the
viable subset for second place); voter $j$ has type $t_j = (\alpha_j,
\beta_j, M_j)$. The match score $M_j(c) \in [0,1]$ summarizes voter
$j$'s ideological and identity-based affinity with candidate $c$;
empirically it is proxied at the location level by the
right-of-centre orientation of the polling place
(Section~\ref{subsec:covariates}).
Let $\mathrm{runoff}(\theta) \triangleq \{\text{Fujimori}, \theta\}$
denote the pair of candidates that advance to the runoff in state
$\theta \in \Theta$. Voter $j$'s utility from voting for $c$ in
state $\theta$ is
\begin{equation}\label{eq:utility}
U_j(c, \theta) \;=\;
\underbrace{\alpha_j\,M_j(c)}_{\text{expressive}}
\;+\;
\underbrace{\beta_j\,M_j(c)\,\mathbf{1}\{c \in \mathrm{runoff}(\theta)\}}_{\text{instrumental}}
\;+\; \varepsilon_{jc},
\end{equation}
with $\alpha_j > 0$, $\beta_j \geq 0$, and i.i.d.\ Type-I Extreme
Value taste shocks $\varepsilon_{jc}$. The instrumental term is
interacted with the expressive match $M_j(c)$: learning that $c$ is
viable moves voter~$j$ toward him only if $M_j(c)$ is appreciable,
so a viable but expressively rejected candidate ($M_j(c) \approx 0$)
exerts no instrumental pull. This multiplicative structure lets a
nationally viable candidate gain votes only where he is also
expressively acceptable.
Define the perceived-viability function $q_c : \Delta(\Theta) \to [0, 1]$
that maps a posterior belief over runoff entrants into the perceived
probability that $c$ is in the runoff:
\begin{equation}\label{eq:qcmu}
q_c(\mu) \;=\;
\begin{cases}
1 & c = \text{Fujimori},\\
\mu(c) & c \in \Theta,\\
0 & \text{otherwise.}
\end{cases}
\end{equation}
Under the standard Gumbel assumption on the idiosyncratic utility
shocks \citep[Ch.~3]{Train2009}, voter~$j$'s probability of
choosing candidate~$c$ after observing the signal pair
$s \in S \times S$ admits the closed-form logit expression
\begin{equation}\label{eq:vote-prob}
\mathbb{P}_j(c \mid s) \;=\;
\frac{\exp\!\bigl(M_j(c)\,[\,\alpha_j + \beta_j\,q_c(\mu_j^{s})\,]\bigr)}
{\sum_{c' \in \mathcal{C}} \exp\!\bigl(M_j(c')\,[\,\alpha_j+ \beta_j\,q_{c'}(\mu_j^{s})\,]\bigr)},
\end{equation}
where $\mu_j^{s} \coloneqq \mu_j(\,\cdot \mid s\,) \in \Delta(\Theta)$
is the posterior of Proposition~\ref{prop:joint-post}. The signal
enters \eqref{eq:vote-prob} only through the viability terms
$q_c(\mu_j^{s})$, but its impact on $c$'s choice probability is
scaled by the expressive match $M_j(c)$: the same posterior shift
moves a voter toward $c$ in proportion to how acceptable $c$ already
is to her. Because the 2026 calls are discordant
($s_D^* \neq s_I^*$), the posterior of
Proposition~\ref{prop:joint-post} splits viability mass between
$s_D^*$ and $s_I^*$ rather than concentrating it on a single
candidate: each named candidate gains in viability, but by strictly
less than under a concordant call.
The behavioral content of \eqref{eq:vote-prob} is a
viability--affinity interaction: a rise in $q_c(\mu_j^{s})$ raises
$\mathbb{P}_j(c \mid s)$ by an amount increasing in $M_j(c)$. The more a
voter already favors $c$, the more a gain in his viability moves her
toward him; where affinity is low, the same gain barely shifts her. The
effect on L\'opez Aliaga and S\'anchez is therefore symmetric: each
gains in proportion to local affinity---L\'opez Aliaga where the
electorate leans right, S\'anchez where it leans left---and neither
draws voters from the other side. The \emph{mal menor} logic of
Section~\ref{sec:parties} reallocates support along existing affinities,
not across them.
\subsection{From individual choices to \emph{acta}-level outcomes}\label{subsec:bridge}
The structural model of Section~\ref{sec:model} operates on the
individual voter $j$, who casts a vote with logit
probability~\eqref{eq:vote-prob}. The data are silent at this level: the
ONPE \textit{actas} record only vote counts at each polling table, and
the secret ballot precludes any link between voters and votes. The
empirical strategy must therefore identify the model-implied effect from
table-level shares alone, which this subsection makes precise.
Voting locations are indexed by $\ell$ and \emph{actas} by $a$, with
$a \in \ell$ denoting that \emph{acta} $a$ is recorded at location $\ell$. Let
$T_\ell \in \{0,1\}$ be the treatment indicator, equal to $1$ if
location $\ell$ voted on Monday 13 April 2026---after the flash
estimates were broadcast---and $0$ otherwise; each \emph{acta} inherits its
location's status, so $T_a \equiv T_\ell$ for $a \in \ell$. The treated
and control locations are
\[
\mathcal{T} \;\coloneqq\; \left\{\, \ell : T_\ell = 1 \,\right\}
\;=\; \left\{\, \ell_1, \ldots, \ell_{13} \,\right\},
\qquad
\mathcal{C}_0 \;\coloneqq\; \left\{\, \ell : T_\ell = 0 \,\right\},
\quad \left| \mathcal{C}_0 \right| \approx 340.
\]
For the $i$-th treated location ($i = 1, \ldots, 13$), let
$A_{\ell_i} = \left\{\, a_{\ell_i}^{1}, \ldots, a_{\ell_i}^{n_i} \,\right\}$
be its set of \emph{actas}, with $n_i = \left| A_{\ell_i} \right|$. The total
number of treated \emph{actas} is
\[
N_T \;\coloneqq\; \sum_{i=1}^{13} n_i \;=\; 187.
\]
At acta $a$ with $N_a$ valid votes, let $V_{jc} = \mathbf{1}\{j
\text{ votes for } c\}$ for voter $j \in a$ and candidate $c$. The
observed share is
\[
Y_{ac} \;=\; \frac{1}{N_a} \sum_{j \in a} V_{jc}.
\]
Under~\eqref{eq:vote-prob}, $\mathbb{E}[V_{jc} \mid \mu_j] =
\mathbb{P}_j(c \mid \mu_j)$. Write $\mu_j^{(0)}$ for voter $j$'s
baseline posterior and $\mu_j^{(1)} \equiv \mu_j(\,\cdot \mid s_D^*,
s_I^*)$ for her posterior after observing the two flash signals
(Proposition~\ref{prop:joint-post}). The expected potential outcome
of \emph{acta} $a$ under treatment $d \in \{0, 1\}$ is
\[
\mathbb{E}[Y_{ac}(d)]
\;=\; \frac{1}{N_a} \sum_{j \in a}
\mathbb{P}_j\left(c \mid \mu_j^{(d)}\right).
\]
Let
\[
\delta_{jc}
\;\coloneqq\; \mathbb{P}_j\left(c \mid \mu_j^{(1)}\right)
- \mathbb{P}_j\bigl(c \mid \mu_j^{(0)}\bigr)
\]
denote the individual information effect for voter $j$ on
candidate $c$. Differencing the potential outcomes yields the
\emph{acta}-level treatment effect as the within-\emph{acta} average of the
$\delta_{jc}$:
\begin{equation}\label{eq:tauac}
\tau_{ac}
\;\coloneqq\; \mathbb{E}[Y_{ac}(1) - Y_{ac}(0)]
\;=\; \frac{1}{N_a} \sum_{j \in a} \delta_{jc}.
\end{equation}
The $\delta_{jc}$ are not observable; their within-\emph{acta} average
$\tau_{ac}$ is the smallest object the model speaks to that is
also visible in the data.
The estimand is the \emph{acta}-weighted average of $\tau_{ac}$ across
the treated population,
\begin{equation}\label{eq:att}
\tau_c \;\coloneqq\;
\frac{1}{N_T} \sum_{i=1}^{13} \sum_{a \in A_{\ell_i}}
\mathbb{E}\bigl[Y_{ac}(1) - Y_{ac}(0) \,\big|\, T_\ell = 1\bigr],
\end{equation}
which is the average treatment effect on the treated (ATT) for candidate
$c$. Its model-implied counterpart is
\begin{equation}\label{eq:ate}
\Delta_c \;\coloneqq\;
\mathbb{E}\!\left[
\frac{1}{N_a} \sum_{j \in a} \delta_{jc}
\;\Big|\; T_\ell = 1
\right]
\;=\;
\mathbb{E}\!\left[
\int \delta_{jc}\, dF_a(j)
\;\Big|\; T_\ell = 1
\right],
\end{equation}
where $F_a$ is the within-\emph{acta} empirical distribution of voter
types (the probability measure that places mass $1/N_a$ on each
voter $j \in a$, so that
$$
\int g(j)\, dF_a(j) = N_a^{-1} \sum_{j
\in a} g(j)
$$
for any function $g$ of voter characteristics).
Equation~\eqref{eq:ate} is the population analogue of the
table-level effect computable in closed form from the structural
model; equation~\eqref{eq:att} is its causal counterpart in the
potential-outcomes framework. The two are linked by the
identification result below.
\begin{assumption}[Selection on observables and overlap]\label{ass:exog}
There exists a vector $H_a$ of pre-treatment covariates---combining
\emph{acta}-level demographics and location-level structural variables,
detailed in Section~\ref{subsec:covariates}---such that, for each
candidate $c \in \mathcal{C}$,
\begin{equation}\label{eq:soo}
Y_{ac}(0) \;\perp\!\!\!\perp\; T_\ell \mid H_a,
\qquad
0 \;<\; \mathbb{P}(T_\ell = 1 \mid H_a) \;<\; 1.
\end{equation}
\end{assumption}
The institutional ground for Assumption~\ref{ass:exog} is the
contractor's logistical protocol of 12--13 April 2026: which
locations were displaced was decided on operational grounds
(transport routes, warehouse-to-centre delivery times, the
operational profile of the districts served) that are correlated
with district-level socioeconomic conditions but orthogonal, by
design, to the political preferences of the location's electorate.
No political variable enters the contractor's protocol. Conditional
on the socioeconomic and demographic covariates that drive the
operational protocol, the residual variation in $T_\ell$ is
therefore as good as random.
\begin{assumption}[Stable Unit Treatment Value Assumption (SUTVA) at the location level]\label{ass:sutva}
The potential outcomes $\bigl(Y_{ac}(0), Y_{ac}(1)\bigr)$ of \emph{acta}
$a \in A_\ell$ depend only on $T_\ell$, not on $T_{\ell'}$ for any
$\ell' \neq \ell$.
\end{assumption}
Assumption~\ref{ass:sutva} rules out between-location spillovers.
\begin{proposition}[Identification of $\tau_c$]\label{prop:identification}
Under Assumptions~\ref{ass:exog} and~\ref{ass:sutva},
\begin{equation}\label{eq:tau-identified}
\tau_c
\;=\; \mathbb{E}\!\left[Y_{ac} \mid T_\ell = 1\right]
\;-\; \mathbb{E}\!\left[m_0(H_a) \mid T_\ell = 1\right]
\;=\; \Delta_c,
\end{equation}
where $m_0(h) \coloneqq \mathbb{E}\!\left[Y_{ac} \mid H_a = h,\, T_\ell = 0\right]$
and $\Delta_c$ is the model-implied effect~\eqref{eq:ate}.\footnote{
$m_0(h)$ is the average vote share for $c$ among control \emph{actas} whose
covariate profile equals $h$. The subtracted term
$\mathbb{E}\!\left[m_0(H_a) \mid T_\ell = 1\right]$ replaces each treated
\emph{acta}'s unobserved untreated outcome with this control average evaluated
at the \emph{acta}'s own covariates; $\tau_c$ is therefore the observed mean
for the treated minus the counterfactual mean they would have recorded
had they voted before the flash. Also, $\mathbb{E}[\cdot|T_{\ell}=1]$ defines the population weighting $1/N_T$.}
\end{proposition}
\begin{proof}
The estimand $\tau_c$ is the average treatment effect on the
treated for candidate $c$, a population mean restricted to treated
\emph{actas}. It is causal and not directly observable. We link it to two
equivalent objects: the model-implied effect $\Delta_c$, computable
in closed form from the structural choice model, and an observable
expression in terms of the control regression function $m_0$. The
argument forms the following triangle:
\begin{center}
\begin{tikzpicture}[
line cap=round,
vrtx/.style ={font=\normalsize, inner sep=2pt},
elbl/.style ={font=\footnotesize, fill=white, inner sep=2pt, rounded corners=1pt},
edgeI/.style ={thick, draw=blue!60!black},
edgeII/.style={thick, draw=red!65!black},
edgeIII/.style={thick, dashed, draw=black!55}
]
\node[vrtx] (tau) at (0, 3) {$\tau_c$};
\node[vrtx, anchor=east] (delta) at (-3.4, 0) {$\Delta_c$};
\node[vrtx, anchor=west] (obs) at (3.4, 0)
{$\mathbb{E}[Y_{ac} \mid T_\ell{=}1] \;-\; \mathbb{E}[m_0(H_a) \mid T_\ell{=}1]$};
\draw[edgeI] (tau) -- (delta)
node[elbl, midway, sloped, above=1pt, text=blue!60!black]
{(I) structural identity};
\draw[edgeII] (tau) -- (obs)
node[elbl, midway, sloped, above=1pt, text=red!65!black]
{(II) population identification};
\draw[edgeIII] (delta) -- (obs)
node[elbl, midway, below=2pt, text=black!60]
{identifying equation};
\end{tikzpicture}
\end{center}
We establish the two upper edges; the bottom edge
is~\eqref{eq:tau-identified} and follows by transitivity.
\medskip
\noindent (I) Structural identity, $\tau_c = \Delta_c$. By
construction of the structural model, the potential vote shares of
\emph{acta} $a$ are within-\emph{acta} averages of the corresponding individual
choice probabilities,
\[
Y_{ac}(0)
\;=\; \frac{1}{N_a} \sum_{j \in a} \mathbb{P}_j\left(c \mid \mu_j^{(0)}\right),
\qquad
Y_{ac}(1)
\;=\; \frac{1}{N_a} \sum_{j \in a} \mathbb{P}_j\left(c \mid \mu_j^{(1)}\right).
\]
Differencing recovers~\eqref{eq:tauac},
$$
\tau_{ac} = N_a^{-1}
\sum_{j \in a} \delta_{jc} = \int \delta_{jc}\, dF_a(j),
$$
where
$F_a$ is the within-\emph{acta} empirical distribution of~\eqref{eq:ate}.
Linearity of conditional expectation, applied to the population of
treated \emph{actas}, then gives
\[
\tau_c
\;=\; \mathbb{E}[\tau_{ac} \mid T_\ell = 1]
\;=\; \mathbb{E}\!\left[
\int \delta_{jc}\, dF_a(j) \;\Big|\; T_\ell = 1
\right]
\;=\; \Delta_c,
\]
which is edge (I).
\medskip
\noindent (II) Population identification. Setting the structural
model aside, the argument is now in the potential-outcomes
framework. Consistency stipulates that the observed share coincides
with the potential share corresponding to the realized treatment,
\[
Y_{ac}
\;=\; T_\ell\, Y_{ac}(1) \;+\; (1 - T_\ell)\, Y_{ac}(0),
\]
so the treated factual mean is identified directly:
\[
\mathbb{E}[Y_{ac}(1) \mid T_\ell = 1]
\;=\; \mathbb{E}[Y_{ac} \mid T_\ell = 1].
\]
The remaining object is the treated counterfactual mean
$\mathbb{E}[Y_{ac}(0) \mid T_\ell = 1]$, the average vote share treated
\emph{actas} would have produced absent the flash. Assumption~\ref{ass:exog}
supplies conditional ignorability,
\begin{equation}\label{eq:ign}
Y_{ac}(0) \;\perp\!\!\!\perp\; T_\ell \mid H_a,
\end{equation}
and overlap, $\mathbb{P}(T_\ell = 0 \mid H_a = h) > 0$ for almost every
covariate profile $h$ attained among treated \emph{actas}, which guarantees a
control comparison group for each such profile. Together they give, for
almost every such $h$,
\begin{align}
\mathbb{E}[Y_{ac}(0) \mid H_a = h, T_\ell = 1]
&\;\stackrel{(a)}{=}\;
\mathbb{E}[Y_{ac}(0) \mid H_a = h, T_\ell = 0]
\notag \\
&\;\stackrel{(b)}{=}\;
\mathbb{E}[Y_{ac} \mid H_a = h, T_\ell = 0]
\;\stackrel{(c)}{=}\; m_0(h),
\label{eq:abc}
\end{align}
where (a) is ignorability~\eqref{eq:ign}, (b) is consistency ($Y_{ac} = Y_{ac}(0)$ when $T_\ell = 0$), and (c) is the
definition of $m_0$. Averaging~\eqref{eq:abc} over the covariate
distribution of the treated (by the tower property) identifies the
counterfactual mean:
\begin{equation}\label{eq:tower}
\mathbb{E}[Y_{ac}(0) \mid T_\ell = 1]
\;=\; \mathbb{E}[m_0(H_a) \mid T_\ell = 1].
\end{equation}
Edge (I) equates the structural and causal effects, $\tau_c = \Delta_c$.
Edge (II) differences the treated factual mean with the
counterfactual~\eqref{eq:tower},
\[
\tau_c
\;=\; \mathbb{E}[Y_{ac} \mid T_\ell = 1]
\;-\; \mathbb{E}[m_0(H_a) \mid T_\ell = 1],
\]
the observable expression estimated in Section~\ref{sec:method}.
Together they close the triangle and yield~\eqref{eq:tau-identified}.\qedhere
\end{proof}
Two readings of~\eqref{eq:tau-identified} matter for what follows. The
identity $\tau_c = \Delta_c$ ties the empirical target to the model and
licenses the sign predictions of Section~\ref{sec:hypotheses}, while the
difference between the treated mean of $Y_{ac}$ and the control-based
counterfactual is the estimable object the matching procedure of
Section~\ref{sec:method} constructs.
\subsection{Treatment as a change of experiment}\label{subsec:bridge-id}
Proposition~\ref{prop:identification} identifies $\tau_c$ as a causal ATT
but not why it is an information effect. The reason: treatment is a
change of \citet{Blackwell1953} experiment over the state space $\Theta$
of Section~\ref{subsec:bayes-one}. A Sunday voter ($T_\ell = 0$) decides
before the flash, under the null experiment $\pi^{\varnothing}$, whose
single signal leaves the posterior at the prior,
$\mu(\,\cdot \mid \pi^{\varnothing}) = \mu_0$. A Monday voter
($T_\ell = 1$) decides after it, under the flash experiment
\begin{equation*}
\pi^{\mathrm{F}} \coloneqq \pi^{\eta_D} \otimes \pi^{\eta_I},
\qquad
\pi^{\mathrm{F}}(s_D, s_I \mid \theta)
= \pi^{\eta_D}(s_D \mid \theta)\,\pi^{\eta_I}(s_I \mid \theta),
\end{equation*}
the conditionally independent (Assumption~\ref{ass:cond-indep})
combination of the Datum and Ipsos kernels, with realized value
$(s_D^*, s_I^*) = (\mathrm{RLA}, \mathrm{RS})$ inducing the posterior
$\mu_i^{(1)}$ of Proposition~\ref{prop:joint-post}. The null discards its
signal---it is a garbling of every experiment---so any experiment
Blackwell-dominates it; hence
$\pi^{\mathrm{F}} \;\succeq_B\; \pi^{\varnothing}$,
strictly once at least one flash kernel is informative, i.e.
$\eta_D, \eta_I < 1 - 1/K$. Treatment thus moves voters, comparable in
$H_a$ by Assumption~\ref{ass:exog}, from $\pi^{\varnothing}$ to
$\pi^{\mathrm{F}}$, and $\tau_c$ is the population behavioral value of
that improvement. Three results stack into
\begin{equation}\label{eq:id-chain}
\begin{aligned}
&\underbrace{\pi^{\mathrm{F}} \;\succeq_B\; \pi^{\varnothing}}_{\text{information design}}
\;\Longrightarrow\;
\underbrace{\delta_{jc} = \mathbb{P}_j\left(c \mid \mu_j^{(1)}\right) - \mathbb{P}_j\left(c \mid \mu_j^{(0)}\right)}_{\text{updating / choice}}
\\[6pt]
&\qquad\Longrightarrow\;
\underbrace{\tau_c = \mathbb{E}\!\left[\textstyle\int \delta_{jc}\, dF_a(j) \,\middle|\, T_\ell = 1\right] = \Delta_c}_{\text{structural estimand}}
\;\overset{\text{Prop.~\ref{prop:identification}}}{=}\;
\underbrace{\mathbb{E}[Y_{ac} \mid T_\ell = 1] - \mathbb{E}[m_0(H_a) \mid T_\ell = 1]}_{\text{matching estimator}},
\end{aligned}
\end{equation}
whose right-hand side Section~\ref{sec:method} estimates.
\begin{remark}[The sign of the effect is the information-design content]\label{rem:id-sign}
A Blackwell improvement makes the finer experiment weakly more valuable
to a Bayesian voter but says nothing about which $c$ gains. The
direction comes from the updating arrow: which posteriors move
and, through the interaction in~\eqref{eq:vote-prob}, which choice
probabilities rise. Hence the hypotheses of Section~\ref{sec:hypotheses}
concern $\mathrm{sign}\,\tau_c$, and national viability
(experiment-level) can diverge from local gain (updating-level), as
the S\'anchez case shows.
\end{remark}
\section{Hypotheses}\label{sec:hypotheses}
The model delivers five sign predictions for the \emph{acta}-level treatment
effects $\tau_c$ defined in~\eqref{eq:att}, all stated for the treated
sample, which is composed of polling stations located in \textit{Lima Sur}. The flash raises
perceived viability $q_c(\mu_j^s)$ above the prior for three
candidates---Nieto (within a percentage point of both named candidates
in either estimate), López Aliaga (Datum's named candidate), and
Sánchez (Ipsos's named candidate)---confirms Fujimori in every state,
and leaves the rest non-viable. The basic prediction is that the
viable candidates gain and the non-viable candidates lose.
By the viability--affinity interaction in~\eqref{eq:vote-prob}, a rise
in $q_c(\mu_j^s)$ moves voter $j$ toward $c$ in proportion to his
expressive match $M_j(c)$, so within the right-leaning Lima sample the
gains rank by match, not by viability alone. Nieto---off-axis, less
extreme, a clean alternative to both blocks---has the highest match;
López Aliaga is right-aligned but tied to the discredited congressional
pact \citep{Infobae2026LopezAliagaCae}, which lowers his; and Sánchez,
left-anchored with a provincial base \citep{Martinelli2026archipielago},
is expressively distant in Lima. The viable gains therefore rank theoretically
Nieto $>$ López Aliaga $>$ Sánchez.
\begin{hypothesis}[Nieto gains the most]\label{hyp:nieto}
$\tau_{\mathrm{JN}} > 0$ and largest among the viable candidates.
\end{hypothesis}
\begin{hypothesis}[López Aliaga gains an intermediate share]\label{hyp:rla}
$0 < \tau_{\mathrm{RLA}} < \tau_{\mathrm{JN}}$, with
$\tau_{\mathrm{RLA}} \geq \tau_{\mathrm{RS}}$.
\end{hypothesis}
\begin{hypothesis}[Sánchez gains least in Lima]\label{hyp:sanchez}
$\tau_{\mathrm{RS}} > 0$ but smallest among the viable candidates:
despite his high national viability, his low Lima popularity
\citep{Martinelli2026archipielago} mutes the local gain.
\end{hypothesis}
\begin{hypothesis}[Fujimori is unmoved]\label{hyp:keiko}
$\tau_{\mathrm{KF}} \approx 0$ (at most weakly positive). Her runoff
entry is confirmed in every state, so the flash conveys no new
viability about her, and strategic voting does not flow to an
already-secured front-runner.
\end{hypothesis}
\begin{hypothesis}[Non-viable candidates lose]\label{hyp:nonviable}
$\tau_c \leq 0$ for every $c \in \mathcal{C}_{\mathrm{nv}}$, where
\[
\mathcal{C}_{\mathrm{nv}}
\;\coloneqq\; \mathcal{C} \setminus
\bigl( \Theta \cup \{\mathrm{KF}\} \bigr).
\]
The clearest losses fall on Álvarez and López Chau; Belmont also
declines, though his election-night withdrawal makes the sign less
determinate.
\end{hypothesis}
The aggregation constraint $\sum_{c \in \mathcal{C}} Y_{ac} = 1$
implies $\sum_{c \in \mathcal{C}} \tau_c = 0$, so the nonnegative
effects in H\ref{hyp:nieto}--H\ref{hyp:keiko} are offset by the losses
in H\ref{hyp:nonviable}; the leading candidates need not net to zero
individually, since the remainder falls on the long tail of minor
candidates.
\section{Empirical strategy}\label{sec:method}
Equation~\eqref{eq:tau-identified} reduces identification to
constructing a credible counterfactual mean for the treated
\emph{actas}. With only thirteen treated locations, a propensity-score
model is fit on too few treated units to be reliable---the estimated
scores are noisy and produce extreme weights---and treatment varies
only at the location level. We therefore match at that level:
cardinality matching selects the largest set of control locations
whose average covariates equal those of the treated locations, up to
preset tolerances, on the covariate vector $H_a$
\citep{ZubizarretaParedesRosenbaum2014, ViscontiZubizarreta2018}.
Unlike the propensity score, it imposes covariate balance directly, as
an explicit constraint, rather than relying on a fitted
treatment-probability model. On the matched sample we estimate
$\tau_c$ two ways: a within-group difference-in-means estimator, and a
within-group regression estimator that additionally corrects for any
covariate differences that remain between treated and control
\emph{actas} within a matched group \citep{PageLenardKeele2020}.
\subsection{Data}\label{sec:data}
The analysis sample covers Lima Sur and comprises 354 voting
locations (\emph{locales de votaci\'on}) and 4930 \emph{actas}.
Of these locations, thirteen are treated and 341 belong to the
control pool. The dataset is built from two primary sources: the
Instituto Nacional de Estad\'istica e Inform\'atica (INEI) and the
Oficina Nacional de Procesos Electorales (ONPE).
\subsubsection{Data sources}\label{subsec:data-sources}
The first source is INEI, which provides georeferenced census
information at the block (\emph{manzana}) level. These data are used
to characterize the socioeconomic and demographic conditions of each
voting location's surroundings, including total population, recent
in-migrants, linguistic composition, private health-insurance
coverage, educational attainment, literacy, ethnic self-identification,
and indicators of access to basic services and housing conditions.
The second source is ONPE, which provides the electoral results of
the 2021 and 2026 general elections, as well as information from the
2026 electoral roll. These data supply the electoral outcomes, the
prior-election variables used to characterize historical voting
behavior, and the demographic composition of registered voters at each
location.
\subsubsection{Construction of location-level variables}
\label{subsec:data-construction}
Census variables are linked to each voting location through a spatial
procedure. For each voting location $\ell$, we construct a
1{,}000-meter (1\,km) area of influence---a buffer---around the polling
place and identify the census blocks that fall within it. Let
$B(\ell)$ denote the set of census blocks within this buffer and let
$\mathrm{pop}_b$ be the population of block $b$. For any block-level
census variable $v_b$, the corresponding location-level measure is
computed as the population-weighted average
\[
V_\ell
\;=\;
\frac{\sum_{b \in B(\ell)} \mathrm{pop}_b\, v_b}
{\sum_{b \in B(\ell)} \mathrm{pop}_b}.
\]
The resulting variables describe the population residing in the
immediate surroundings of each voting location rather than the average
conditions of the administrative district.
This spatial criterion is also consistent with the institutional logic
of voter-location assignment in Peru. Through the \emph{Elige tu local}
program, voters are allowed to select up to three preferred polling
locations, typically based on proximity to their residence, and to rank
them according to their desired priority. The final assignment depends
on the availability of capacity at the selected locations, but the
mechanism nevertheless creates a direct link between voting locations
and the residential areas surrounding them.
The 2021 electoral results are then linked to the voting locations
used in the analysis. Where a voting location can be tracked across
electoral processes, the correspondence is made directly. Where no
exact match exists, the assignment relies on geographic proximity and
similarity in the number of polling tables (\emph{mesas de votaci\'on}),
so as to identify the most comparable prior-election location.
\subsection{Covariate vector}\label{subsec:covariates}
Matching uses nine location-level covariates, all measured before
13 April 2026 and constant across a location's \emph{actas}. We index
these covariates by the \emph{acta} $a$ under the convention
$H_a = H_{\ell(a)}$, since the balance constraints~\eqref{eq:cm} act
on \emph{acta}-weighted means. The covariates fall into three blocks.
The first block contains four socioeconomic variables constructed
from INEI block-level census data using the spatial procedure described
above:
\[
V \in
\{\mathrm{SES}, \mathrm{Indig}, \mathrm{EDU}, \mathrm{Net}\}.
\]
These variables are the socioeconomic stratum
$\mathrm{SES}_\ell$, measured by the INEI stratum index from $1$
lowest to $5$ highest; the indigenous-mother-tongue share
$\mathrm{Indig}_\ell$; the completed-higher-education share
$\mathrm{EDU}_\ell$; and the internet-access share
$\mathrm{Net}_\ell$. The last three variables lie in $[0,1]$.
The second block summarizes prior political orientation using 2021
first-round vote shares. Let $s_{\ell,c}^{2021}$ be candidate $c$'s
share of the valid vote at location $\ell$ in the first round of the
2021 general election, and let $\mathcal{L}^{2021}$,
$\mathcal{R}^{2021}$, and $\mathcal{F}^{2021}$ be the disjoint sets
\[
\mathcal{L}^{2021} = \{\text{Per\'u Libre, Juntos por el Per\'u, Partido Morado, Acci\'on Popular}\},
\]
\[
\mathcal{R}^{2021} = \{\text{Avanza Pa\'is, Renovaci\'on Popular}\}, \qquad
\mathcal{F}^{2021} = \{\text{Fuerza Popular}\}.
\]
The three political-orientation variables are
\[
\mathrm{Left}_\ell^{2021}
=
\sum_{c \in \mathcal{L}^{2021}} s_{\ell,c}^{2021},
\qquad
\mathrm{Right}_\ell^{2021}
=
\sum_{c \in \mathcal{R}^{2021}} s_{\ell,c}^{2021},
\qquad
\mathrm{Fuji}_\ell^{2021}
=
\sum_{c \in \mathcal{F}^{2021}} s_{\ell,c}^{2021}.
\]
All three variables lie in $[0,1]$. Splitting prior orientation into
three blocs rather than imposing one balance constraint per 2021
candidate is required by the narrow treated sample, which cannot
support a separate balance constraint for each candidate
under~\eqref{eq:cm}. Because the three sets are disjoint, the shares do
not double-count votes.
The third block comes from the 2026 electoral roll at the location:
the share of female registered voters
$\mathrm{Fem}_\ell \in [0,1]$ and the mean registered-voter age
$\mathrm{Age}_\ell$, standardized across locations to have mean $0$
and unit variance. The resulting balance vector is
\begin{equation}\label{eq:Hal}
H_a =
\bigl(
\mathrm{SES}_\ell,\;
\mathrm{Indig}_\ell,\;
\mathrm{EDU}_\ell,\;
\mathrm{Net}_\ell,\;
\mathrm{Left}_\ell^{2021},\;
\mathrm{Right}_\ell^{2021},\;
\mathrm{Fuji}_\ell^{2021},\;
\mathrm{Fem}_\ell,\;
\mathrm{Age}_\ell
\bigr).
\end{equation}
Matching is conducted within district. The district marginal is imposed
on~\eqref{eq:cm} as a fine-balance side constraint. The cardinality step
selects $\mathcal{C}^\star$ from the initial control pool
$\mathcal{C}_0$, and the second stage assigns each selected control to
a within-district treated location by Mahalanobis distance, forming the
matched groups $G_i$ described in Section~\ref{subsec:cardinality}.
\subsection{Descriptive statistics}\label{subsec:descriptives}
Table~\ref{tab:descriptives} reports the raw, pre-matching distribution
of the balance covariates and the 2026 outcomes for the thirteen
treated and the 341 control locations. Moments are
\emph{acta}-weighted, so each location enters in proportion to its
number of \emph{actas}, consistent with the \emph{acta}-weighted
estimand $\tau_c$ in~\eqref{eq:att}. The reported $N$ counts voting
locations, which is the level at which treatment is assigned and at
which the covariates are constant.
Two patterns stand out. First, treated locations sit below the control
pool on the socioeconomic gradient: they have a lower socioeconomic
stratum ($2.21$ vs.\ $2.60$), lower completed-higher-education share
($0.249$ vs.\ $0.312$), lower internet access ($0.361$ vs.\ $0.453$),
and a higher indigenous-language share ($0.110$ vs.\ $0.091$). This is
precisely the imbalance that the matching procedure in
Section~\ref{subsec:cardinality} is designed to remove. Second,
political-orientation covariates are already relatively close before
matching: the 2021 left, right, and Fujimori shares differ by at most
2.2 percentage points, and the female-voter share is nearly identical
across groups ($0.492$ vs.\ $0.499$). This pattern is consistent with
Assumption~\ref{ass:exog}: displacement is related to the socioeconomic
profile of the served areas, rather than to their prior electoral
preferences.
The treated group is also tighter, with smaller standard deviations on
every covariate, and its support lies inside the control range
throughout. Thus, common support holds before matching and the matched
controls are drawn by interpolation rather than extrapolation.
Panel~B reports the seven leading candidates' 2026 first-round vote
shares, the outcomes $Y_{ac}$. The role of these gaps differs from that
of the covariate gaps in Panel~A. Covariate imbalance is a nuisance that
the matching design removes, whereas outcome gaps are the quantities of
interest. If the overnight Ipsos and Datum estimates affected Monday
ballots, treated--control differences in vote shares are precisely what
the design should detect.
The raw outcome contrasts are sizable. Treated locations show higher
mean shares for Fujimori ($0.236$ vs.\ $0.201$), S\'anchez
($0.064$ vs.\ $0.031$), L\'opez Aliaga ($0.186$ vs.\ $0.161$), and
especially Nieto ($0.311$ vs.\ $0.148$). They show lower shares for
Belmont ($0.080$ vs.\ $0.098$), \'Alvarez ($0.042$ vs.\ $0.106$), and
L\'opez Chau ($0.015$ vs.\ $0.060$). These raw contrasts, however, do
not by themselves identify the treatment effect because they still
combine the Monday treatment with the socioeconomic imbalance described
in Panel~A. The matched design of Section~\ref{subsec:cardinality}
isolates the treatment component, with estimates and interpretation
reported in Section~\ref{sec:results}.
\subsection{Notation for the matched design}\label{subsec:matched-notation}
The empirical procedure returns, in two stages
(Section~\ref{subsec:cardinality}), a control sample
$\mathcal{C}^\star \subseteq \mathcal{C}_0$ and a partition of
$\mathcal{T} \cup \mathcal{C}^\star$ into thirteen matched groups
\[
G_i \;=\;
\left\{ \ell_i,\, \tilde\ell_1^{\,i},\, \tilde\ell_2^{\,i},\,
\ldots,\, \tilde\ell_{p_i}^{\,i} \right\},
\qquad i = 1, \ldots, 13,
\]
where $\tilde\ell_r^{\,i} \in \mathcal{C}^\star$ is the $r$-th
control location matched to the treated location $\ell_i$, and $p_i$
is the number of controls in group $G_i$. Group sizes are not fixed:
$p_i$ is determined endogenously by the matching and constrained to
lie between three and ten, $3 \le p_i \le 10$. The lower bound rules
out groups so small that a single atypical control would drive the
within-group contrast; the upper bound keeps a treated stratum from
being diluted by a large, internally heterogeneous control set and
preserves the locality of the comparison. Inside group
$G_i$, the \emph{actas} of the $r$-th control are
\[
A_{\tilde\ell_r^{\,i}}
\;=\; \left\{ a_{\tilde\ell_r^{\,i}}^{\,1}, \ldots,
a_{\tilde\ell_r^{\,i}}^{\,n_{\tilde\ell_r^{\,i}}}
\right\},
\]
and the total number of control \emph{actas} in $G_i$ is $N_i^C \coloneqq
\sum_{r=1}^{p_i} n_{\tilde\ell_r^{\,i}}$. Aggregating across groups,
$N_C \coloneqq \sum_{i=1}^{13} N_i^C$ is the size of the matched
control sample at the \emph{acta} level.
The outcome at \emph{acta} $a \in A_{\ell_i} \cup \bigcup_{r}
A_{\tilde\ell_r^{\,i}}$ is the share $Y_{ac}$ for candidate $c$.
Treatment is constant within group: every \emph{acta} in $A_{\ell_i}$ has
$T_\ell = 1$ and every \emph{acta} in $\bigcup_{r}
A_{\tilde\ell_r^{\,i}}$ has $T_\ell = 0$. Each treated location
therefore defines its own matched stratum, and identification
within $G_i$ is the comparison between treated \emph{actas} at $\ell_i$
and the pooled \emph{actas} of its $p_i$ matched controls. Finally, $n_i \equiv n_{\ell_i} = |A_{\ell_i}|$.
\subsection{Cardinality matching at the location level}\label{subsec:cardinality}
Cardinality matching selects the largest control sample that satisfies
pre-specified balance constraints, rather than minimizing a pairwise
distance \citep{ZubizarretaParedesRosenbaum2014, ViscontiZubizarreta2018}.
With only thirteen treated clusters, propensity-score models are unstable;
fixing the balance target ex ante and maximizing the matched sample
subject to it is more robust. Let $q_\ell \in \{0,1\}$ indicate inclusion
of control location $\ell \in \mathcal{C}_0$. The program maximizes the
matched control sample, measured in \emph{actas}, subject to
\emph{acta}-weighted balance on every component of $H_a$:
\begin{equation}\label{eq:cm}
\max_{\{q_\ell\}} \; \sum_{\ell \in \mathcal{C}_0} q_\ell\, n_\ell
\qquad \text{s.t.} \qquad
\frac{\bigl| \bar H_{j,T} - \bar H_{j,\mathcal{C}^\star} \bigr|}{s_j}
\;\leq\; \delta_j, \qquad j = 1, \ldots, J,
\end{equation}
where $J = \dim(H_a)$, $s_j$ is the pooled standard deviation of
$H_{j,a}$, and the left-hand side is the absolute standardized mean
difference (SMD) of covariate $j$ between the treated sample and the
selected controls. The \emph{acta}-weighted means are
\[
\bar H_{j,T}
= \frac{1}{N_T} \sum_{i=1}^{13} \sum_{a \in A_{\ell_i}} H_{j,a},
\qquad
\bar H_{j,\mathcal{C}^\star}
= \frac{1}{N_{\mathcal{C}^\star}} \sum_{\ell \in \mathcal{C}_0}
q_\ell \sum_{a \in A_\ell} H_{j,a},
\]
with $N_{\mathcal{C}^\star} = \sum_{\ell} q_\ell n_\ell$ and selected
controls $\mathcal{C}^\star = \{\ell : q_\ell = 1\}$. The baseline imposes
a strict balance rule: every covariate must meet $\delta_j = 0.10$, that
is $|\mathrm{SMD}_j| \le 0.10$ for all $j$ with no exceptions. The $0.10$
ceiling is the conventional threshold below which covariate imbalance is
treated as negligible \citep{Austin2009, Stuart2010}; entering it as a
hard, per-covariate constraint---rather than as an average or an
aggregate target---ensures no single covariate is left imbalanced in
exchange for tighter balance elsewhere. Section~\ref{subsec:robustness}
reruns the match at the tighter and looser caps $\delta_j \in \{0.05,
0.15\}$ to confirm the result is not an artifact of the $0.10$ cutoff.
Fine balance on the district marginal enters as a side constraint: a
treated district must contribute the same share of selected controls as
of treated locations. Problem~\eqref{eq:cm} is solved in Python.
Problem~\eqref{eq:cm} selects which control locations enter the sample,
but not which of them serves as a comparison for each treated location.
The estimators below need that assignment: each treated location must
sit in a group with its own comparable controls. A second stage
therefore partitions $\mathcal{T} \cup \mathcal{C}^\star$ into the
thirteen groups $G_i$, one per treated location, with each treated
location receiving between three and ten controls. Controls are assigned by Mahalanobis distance on the continuous
sub-vector $Z_\ell^{\mathrm{cont}} :=(\mathrm{SES}_\ell,\;
\mathrm{EDU}_\ell,\;
\mathrm{Left}_\ell^{2021},\;
\mathrm{Right}_\ell^{2021},\;
\mathrm{Fuji}_\ell^{2021},\;
\mathrm{Fem}_\ell,\;
\mathrm{Age}_\ell)$,
\[
d(\ell, \ell')
= (Z_\ell^{\mathrm{cont}} - Z_{\ell'}^{\mathrm{cont}})^\top
\Sigma_{Z^{\mathrm{cont}}}^{-1}
(Z_\ell^{\mathrm{cont}} - Z_{\ell'}^{\mathrm{cont}}),
\]
with $\Sigma_{Z^{\mathrm{cont}}}$ the empirical covariance of
$Z_\ell^{\mathrm{cont}}$, so that a small $d$ marks two very similar
locations. A control may be assigned to a treated location only if it
lies inside a Mahalanobis caliper, $d(\ell, \ell') \le 3$; controls
beyond it are left unassigned rather than forced into a poorly matched
group. Controls
failing the caliper are dropped from the matched sample, so the balance
reported in Section~\ref{sec:results} is computed on the retained
controls; otherwise the stage only groups the already-balanced controls
into locally homogeneous comparison sets.
\subsection{Two estimators on the matched sample}\label{subsec:estimators}
We estimate $\tau_c$ on the matched groups $\{G_i\}_{i=1}^{13}$ with two
estimators that share this design but differ in how they handle residual
within-group imbalance. The first is a nonparametric difference-in-means
(DM) estimator. The treated and control \emph{acta}-weighted means in
$G_i$ are
\begin{equation}\label{eq:groupmeans}
\bar Y_{i,c}^{\,T}
\;=\; \frac{1}{n_i} \sum_{a \in A_{\ell_i}} Y_{ac},
\qquad
\bar Y_{i,c}^{\,C}
\;=\; \frac{1}{N_i^C} \sum_{r=1}^{p_i}
\sum_{a \in A_{\tilde\ell_r^{\,i}}} Y_{ac},
\end{equation}
with group contrast $\hat\Delta_{i,c} \coloneqq \bar Y_{i,c}^{\,T} -
\bar Y_{i,c}^{\,C}$. The ATT estimator weights each group by its
\emph{acta} share of the treated sample, $w_i \coloneqq n_i / N_T$:
\begin{equation}\label{eq:dim}
\widehat\tau_c^{\,\mathrm{DM}}
\;=\; \sum_{i=1}^{13} w_i\, \hat\Delta_{i,c}.
\end{equation}
These weights reproduce the \emph{acta}-weighted ATT of~\eqref{eq:att}: a
treated location with more \emph{actas} counts proportionally more.
Equation~\eqref{eq:dim} imposes no functional form linking $Y_{ac}$ to
$H_a$; it relies entirely on the balance from~\eqref{eq:cm}. As a
robustness check, Section~\ref{subsec:robustness} also reports the
location-weighted variant (superscript $L$), which weights each treated
location equally regardless of its number of \emph{actas}:
\begin{equation}\label{eq:tauL}
\widehat\tau_c^{\,\mathrm{DM}, L}
\;=\; \frac{1}{13} \sum_{i=1}^{13} \hat\Delta_{i,c}.
\end{equation}
The second estimator adjusts for residual imbalance in the
\emph{acta}-level demographics $X_a$ (age and female participation), following standard practice for
clustered observational studies \citep{PageLenardKeele2020}. Group by
group, run
\begin{equation}\label{eq:reg-group}
Y_{ac} \;=\; \alpha_{i,c} \;+\; \beta_{i,c}\, T_\ell
\;+\; X_a^\top \psi_{i,c} \;+\; u_{ac},
\qquad a \in G_i,
\end{equation}
where $T_\ell = 1$ for $a \in A_{\ell_i}$ and $0$ otherwise, and
$\psi_{i,c}$ is the within-group slope vector on $X_a$.
The location covariates $Z_\ell$ are omitted. They are constant on the treated arm (one treated location per group), but on the control arm $Z_\ell$ varies across the matched controls and is therefore not swept out by the group fixed effect; what makes its omission innocuous is the balance constraint~\eqref{eq:cm}, which holds each treated location's $Z_\ell$ to its matched-control mean within the imposed tolerance, so the residual imbalance in $Z_\ell$ is minimized by design rather than identically absorbed by the fixed effects. The group-by-group aggregate of the within-group treatment effects, with
the same weights,
\begin{equation}\label{eq:reg-agg}
\sum_{i=1}^{13} w_i\, \hat\beta_{i,c},
\end{equation}
uses the group-specific slopes $\psi_{i,c}$ of~\eqref{eq:reg-group}. We
instead compute the more parsimonious pooled, \emph{acta}-weighted
regression
\begin{equation}\label{eq:reg-pool}
Y_{ac} \;=\; \alpha_{g(\ell), c} \;+\; \tau_c\, T_\ell
\;+\; X_a^\top \psi_c \;+\; \varepsilon_{ac},
\end{equation}
with a matched-group fixed effect $\alpha_{g(\ell), c}$ for the group
containing $\ell$, a single treatment coefficient $\tau_c$, and slopes
$\psi_c$ common across groups, each \emph{acta} weighted so that the
treated and control arms of $G_i$ each carry total weight $w_i$ (each
treated \emph{acta} gets $1/N_T$; each control \emph{acta} in $G_i$ gets
$w_i/N_i^C$); we set $\widehat\tau_c^{\,\mathrm{R}}$ equal to the OLS
estimate of $\tau_c$ in~\eqref{eq:reg-pool}. Dropping $X_a$ recovers
$\widehat\tau_c^{\,\mathrm{DM}}$ exactly. The two regression adjustments
coincide only when the covariate slopes are homogeneous across groups
($\psi_{i,c} = \psi_c$) or $X_a \perp T_\ell$ within every $G_i$; with
heterogeneous slopes \eqref{eq:reg-agg} and~\eqref{eq:reg-pool} differ,
because the latter pools the covariate adjustment across groups while
the former does not.
The two estimators differ only in how they treat residual within-group
imbalance: $\widehat\tau_c^{\,\mathrm{DM}}$ takes the raw contrast,
$\widehat\tau_c^{\,\mathrm{R}}$ adjusts for $X_a$ through the common
slopes $\psi_c$ of~\eqref{eq:reg-pool}. They coincide when $X_a \perp T_\ell$ within $G_i$, and
their gap measures how much the adjustment matters. We report
$\widehat\tau_c^{\,\mathrm{DM}}$ as the headline estimate and
$\widehat\tau_c^{\,\mathrm{R}}$ as the regression-adjusted check.
\subsection{Balance diagnostics}\label{subsec:diagnostics}
The design's validity rests on the matched controls reproducing the
treated covariate distribution. The main diagnostic is the post-match
SMD of each component of $H_a$,
\begin{equation}\label{eq:smd}
\mathrm{SMD}_j
\;=\; \frac{\bar H_{j, T} - \bar H_{j, \mathcal{C}^\star}}{s_j},
\end{equation}
with the practical threshold $|\mathrm{SMD}_j| < 0.10$;
Table~\ref{tab:balance} reports the before- and after-matching SMDs. The
full battery---per-covariate Kolmogorov--Smirnov statistics,
before-and-after density plots of $H_{j,a}$, the geographic spread of
treated and selected control locations, and the matched-sample sizes
$|\mathcal{C}^\star|$ and $N_C$ (which show whether the comparison rests
on a wide control pool or a few influential units)---is reported in
Appendix~\ref{app:balance}.
\subsection{Sensitivity and robustness}\label{subsec:robustness}
Given the small treated sample, the estimate is stress-tested four ways.
\begin{enumerate}
\item Leave-one-treated-location-out: recompute the estimator dropping
each treated location in turn, $\widehat\tau_{c, (-i)}^{\,\mathrm{ATT}}$
for $i = 1, \ldots, 13$. A sign reversal or a sharp change in magnitude
driven by one location would warrant interpretive caution.
\item Location-weighted estimand: $\widehat\tau_c^{\,\mathrm{DM}, L}$
of~\eqref{eq:tauL}, equal weight per treated location.
\item Balance tolerance: resolve~\eqref{eq:cm} for $\delta_j \in
\{0.05, 0.10, 0.15\}$.
\item Rosenbaum bounds: under the matched-group sensitivity framework, the sensitivity parameter
$\Gamma$ bounds how far two matched voting locations with identical
observed covariates could differ in their odds of treatment exposure due
to unobserved factors. Following the matched-group design adopted in this
study, sensitivity is evaluated using an aggregated signed-score
statistic based on the within-group treatment--control contrasts
$\widehat{\Delta}_{i,c}$. We report $\Gamma^\star$, defined as the
largest value of $\Gamma$ for which the sharp null hypothesis of no
treatment effect is still rejected at the $5\%$ significance level
(Table~\ref{tab:rosen}). Higher values of $\Gamma^\star$ indicate that
the estimated treatment effect is more robust to potential unobserved
confounding, whereas values close to one suggest that relatively small
departures from the selection-on-observables assumption would be
sufficient to overturn the inference
\citep{Rosenbaum2002}.
\end{enumerate}
\subsection{Limitations}\label{subsec:limitations}
The main limitation is the small number of treated locations: although
the control pool is large, the independent causal information comes from
$L_T = 13$ clusters, so the estimates are evidence for the displaced
locations rather than a generalizable nationwide effect. Second, the
match balances only observed covariates, so sensitivity to unobserved
confounding remains a concern the design cannot rule out.
\section{Results}\label{sec:results}
\subsection{Reduced-form effects}\label{subsec:reduced-form}
The baseline matched design retains the thirteen treated locations
(187 \emph{actas}) and selects 178 control locations ($2386$
\emph{actas}) by cardinality matching; the within-district Mahalanobis
caliper then retains 116 of these controls ($1722$ \emph{actas}), so the
estimation sample comprises 129 matched locations---13 treated and 116
control---with $1909$ \emph{actas} in total.
Table~\ref{tab:balance} reports the standardized mean difference
(SMD) for each matching covariate before and after matching. The
largest absolute imbalance falls from $0.79$ in the full control
pool to $0.100$ in the matched sample, and every covariate clears
the imposed tolerance, so the treated and matched-control
distributions are comparable on observables.
Table~\ref{tab:att} reports the two estimators of $\tau_c$ for the
seven leading candidates: the \emph{acta}-weighted difference-in-means
estimator $\widehat\tau_c^{\,\mathrm{DM}}$ of~\eqref{eq:dim} and regression-adjusted estimator $\widehat\tau_c^{\,\mathrm{R}}$
of~\eqref{eq:reg-pool} with cluster-robust inference. The two
agree in sign and magnitude for every candidate, so the adjustment
for residual demographic imbalance does little work and the global
balance of~\eqref{eq:cm} carries the identification.
Three patterns emerge. First, the three candidates the flash
estimates rendered viable---Nieto, L\'opez Aliaga, and
S\'anchez---all gain significant vote share among Monday voters
relative to matched Sunday locales, and the leading non-viable
candidates---\'Alvarez, Belmont, and L\'opez Chau---all lose share.
This is the sign pattern the strategic-reallocation logic of the
model predicts (H\ref{hyp:nieto}--H\ref{hyp:sanchez},
H\ref{hyp:nonviable}). Second, the incumbent runoff entrant Fujimori
gains a small but significant share ($+2.6$ percentage points),
consistent with the weak-reinforcement prediction of
H\ref{hyp:keiko}. Third, the reallocation is overwhelmingly
concentrated on Nieto, whose $+17.3$-point gain is three to seven
times the gain of any other candidate. The seven
leading candidates' effects sum to $+0.134$, so the remaining $13.4$
points of displaced Monday share are drawn from the long tail of
minor candidates, consistent with the adding-up implication of
H\ref{hyp:nonviable}.
\subsection{Reading the viability--affinity interaction from the magnitudes}\label{subsec:reading}
The estimated magnitudes track the affinity ranking of
Section~\ref{sec:hypotheses}. The signs reproduce the model's core
division of the ballot---share moves toward the viable candidates
and away from the rest---and the within-viable ordering is
Nieto $>$ L\'opez Aliaga $>$ S\'anchez, exactly as
H\ref{hyp:nieto}--H\ref{hyp:sanchez} anticipate. Nieto, the less
extreme option standing outside the discredited left--right blocs,
combines the viability the flash conferred with the highest
expressive match in the displaced electorate and captures the
largest gain; L\'opez Aliaga, viable and ideologically aligned but
weighed down by his identification with the congressional pact,
gains an intermediate share; S\'anchez, viable nationally but
expressively distant in this right-leaning metropolitan sample,
gains the least. Because H1–H3 are ordinal, the size of the gap is not a departure from the predictions but a quantitative reading of them: Nieto's gain is several times López Aliaga's, which underscores how strongly the off-axis clean attribute operates in the displaced electorate relative to the two right-aligned alternatives. The
aggregate magnitudes are consistent with the viability--affinity
interaction but cannot, on their own, separate affinity from
viability: a candidate's national viability and his local gain need
not coincide, as the S\'anchez case makes clear.
\subsection{Robustness}\label{subsec:results-robustness}
The headline pattern survives every exercise of the sensitivity
battery of Section~\ref{subsec:robustness}.
\paragraph{Location-weighted estimand.} Re-weighting so that each
treated location counts equally, rather than in proportion to its
number of \emph{actas}, leaves every sign and significance unchanged
(Table~\ref{tab:lw}). The point estimates are close to the
\emph{acta}-weighted ones---Nieto $+0.176$, L\'opez Aliaga $+0.054$,
S\'anchez $+0.025$---so the result is not driven by the larger
treated locations.
\paragraph{Leave-one-treated-location-out.} Recomputing the
estimator thirteen times, excluding one treated location at a time,
produces no sign reversal and no loss of significance at the $5\%$
level for any candidate (Table~\ref{tab:loo}). The ranges are tight:
Nieto's effect stays within $[0.170,\,0.178]$, and the group-by-group
decomposition is broad-based---each viable candidate's within-group
contrast is positive in at least $11$ of the $13$ groups and no single
group accounts for more than a quarter of any aggregate effect.
\paragraph{Balance tolerance.} Re-solving the cardinality-matching
problem~\eqref{eq:cm} at $\delta \in \{0.05, 0.10, 0.15\}$ moves the
estimates negligibly (Table~\ref{tab:tol}); the result is not an
artifact of a single tolerance. Tighter tolerances select fewer
cardinality-selected cardinality-selected controls (163, 178, 200 locations) at a better post-match balance, and all estimates remain significant at the $1\%$ level.
\paragraph{Rosenbaum bounds.} Under the matched-group sensitivity framework described in
Section~\ref{sec:method}, Table~\ref{tab:rosen} reports the
worst-case $p$-value associated with each electoral outcome as the
sensitivity parameter $\Gamma$ increases. For each outcome, the
procedure evaluates how large an unobserved source of selection would
need to be to overturn the inference obtained from the matched-group
treatment--control contrasts. The reported value $\Gamma^\star$
corresponds to the largest degree of hidden bias for which the sharp
null hypothesis of no treatment effect continues to be rejected at the
$5\%$ significance level. Larger values of $\Gamma^\star$ indicate
greater robustness to unobserved confounding, as the estimated effect
remains statistically significant even when matched voting locations are
allowed to differ substantially in their odds of treatment exposure due
to factors not captured by the observed covariates. Conversely, values
of $\Gamma^\star$ close to one suggest that relatively small departures
from the selection-on-observables assumption would be sufficient to
eliminate statistical significance.
\textcolor{red}{\textbf{Note:} The Rosenbaum bounds reported in this section are preliminary. The aggregated signed-score statistic is appropriate for matched pairs but does not faithfully characterize sensitivity in $1{:}k$ sets with a varying number of controls: in a set with $p_i$ controls, permuting the treatment assignment yields $1+p_i$ distinct contrasts rather than a sign flip of the average contrast over $\pm\hat\Delta_{i,c}$. The current implementation also decouples the inference base (within-group permutation) from the sensitivity base and does not reproduce the permutation $p$-value at $\Gamma=1$. This will be corrected in the next version with a faithful matched-sets implementation: per-location scores, binary worst-case within-set tilt under $\Gamma$, and exact combination across the 13 independent groups.}
\section{Conclusion}\label{sec:conclusion}
We study how election-night flash estimates reshape voting in a
fragmented plurality election. A Bayesian-updating model in which
voter $j$'s payoff from candidate $c$ is
$\alpha_j M_j(c) + \beta_j\, M_j(c)\,\mathbf{1}\{\cdot\}$ yields a
viability--affinity interaction: a public signal raises perceived
viability $q_c$ uniformly, but the vote it reallocates is scaled by the
expressive match $M_j(c)$, making reallocation a within-neighborhood
phenomenon rather than a mechanical flow toward whoever the flash names.
We exploit a natural experiment---the \emph{Jurado Nacional de
Elecciones} extended voting in 13 Lima locations to the Monday after
election day, exposing 187 \emph{actas} to the overnight Ipsos and Datum
estimates while comparable Sunday locations were not---and estimate
candidate-level effects on 2026 first-round shares by cardinality
matching (129 matched locations: 13 treated and 116 control; 1909 \emph{actas}) with cluster-robust
inference. The acta-weighted difference-in-means and within-group
regression estimators agree in sign and magnitude for every candidate, so the adjustment
for residual demographic imbalance does little work.
Identification, however, rests on within-group comparability rather
than on the pooled balance alone: the estimators weight each group's
control \emph{actas} by $w_i/N_i^C$, which departs from the
equal-\emph{acta} weighting $1/N_{\mathcal{C}^\star}$ of the pooled
SMDs in Table~\ref{tab:balance} once group sizes differ. Balance under
this weighting is secured by the within-district Mahalanobis caliper of
Section~\ref{subsec:cardinality}, which keeps each treated location
close to its own matched controls, while~\eqref{eq:cm} delivers the
global marginal balance.
Our contribution is to identify the causal effect of a late viability
signal on vote reallocation and to show that it runs through the local
distribution of expressive affinities, not the signal alone; the model
is the organizing device for this claim, not a stand-alone result.
Information has a significant, robust effect. The flash raised the shares
of the three candidates it rendered viable (Nieto, L\'opez Aliaga,
S\'anchez; H\ref{hyp:nieto}--H\ref{hyp:sanchez}), weakly reinforced
Fujimori ($+2.6$ points; H\ref{hyp:keiko}), and lowered the non-viable
candidates' shares (H\ref{hyp:nonviable}); every effect is significant at
$1\%$ and survives the leave-one-out, weighting, tolerance, and
Rosenbaum checks. The
magnitudes follow $M_j(c)$: because $q_c$ moves uniformly while the
reallocation is scaled by match, the ordering
Nieto $>$ L\'opez Aliaga $>$ S\'anchez tracks affinity, not national
viability. Nieto, the off-axis ``clean'' option with the highest match,
takes the bulk ($+17.3$ points); L\'opez Aliaga's match is depressed by
his tie to the congressional pact ($+5.1$); S\'anchez, the eventual
runoff entrant, is expressively most distant in this right-leaning
sample and gains least ($+2.7$). A candidate can thus advance nationally
on a signal yet move few local votes, because viability travels with the
signal while its reallocation is governed by local affinity.
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