Strategic Voting and the Swing Voter's Curse

Strategic Voting and the Swing Voter's Curse

An Assumption We Have Been Making

Throughout the information aggregation lectures, every juror has voted according to her signal. If she received $g$, she voted to convict; if she received $i$, she voted to acquit. This behavior was presented as natural, even obvious. But it was an assumption — not a derived result.

The assumption is that each juror's "information" consists solely of her own signal. She votes as if no other jurors exist. She does not ask: given that I am in a situation where my vote changes the outcome, what does that tell me about what everyone else has seen?

This is the question we now take seriously. And it changes things — but only under certain voting rules.

Strategic Voting in Real Elections

Before turning to the formal model, it is worth observing that voting strategically — taking into account the behavior of other voters rather than simply expressing one's sincere preference — is a real and documented phenomenon.

In the 2024 UK general election, websites like Best for Britain provided constituency-by-constituency advice on which opposition candidate (Labour, Liberal Democrat, or Green) had the best chance of defeating the local Conservative candidate. Millions of voters followed the advice and voted for their second or third choice. The Conservatives lost 250 seats — their worst result since 1906.

In the 2016 US election, Green Party voters in Michigan preferred Jill Stein to Hillary Clinton. Stein received 51,000 votes in that state. Trump won Michigan by 10,700 votes. Voters who preferred Stein but voted for her sincerely, rather than strategically switching to Clinton, may have altered the election's outcome.

In Russia, Alexei Navalny's "Smart Voting" campaign identified the single strongest non-United Russia candidate in each district and urged opposition voters to support that person — regardless of their personal preference for that candidate. The strategy was designed to concentrate opposition votes rather than disperse them across ideologically diverse parties.

All three examples share a common feature: there are at least three alternatives and the voters have conflicting interests. A Green voter does not have the same interests as a Labour voter. The strategic behavior arises because a sincere vote might "waste" influence when the voter's true preference has no chance of winning.

What is more surprising — and what this lecture establishes — is that strategic voting arises even in settings with only two alternatives and a common interest. No third-party spoiler, no value disagreements: just a jury trying to reach the correct verdict, under a voting rule that creates incentives to deviate from sincere behavior.

The Main Results

We can now state the two central results of this lecture, which apply to the $N$-juror model under the assumptions established in Lectures 11–13 (binary signals with accuracy $p > \frac{1}{2}$, equal priors, threshold $1-p < q \leq p$):

Under simple majority: sincere voting is rational, even for a strategic juror. The analysis of earlier lectures was without loss of generality — a juror who accounts for the behavior of others still wants to vote according to her own signal.

Under unanimity: sincere voting is not always rational. A strategic juror may want to vote differently from her signal.

The second result is the focus of this lecture. We derive it concretely for $N = 3$.

Pivotality Under Unanimity

Consider a jury of $N = 3$ under unanimity: the defendant is convicted only if all three jurors vote to convict. Assume the other two jurors are sincere — they vote according to their signals. You are strategic: you vote to maximize the probability of the correct outcome, conditioning on all the information available to you.

You receive signal $i$. How do you vote?

Recall the voting rule: you vote to convict if $\text{Prob}(G \mid \text{your information}) \geq q$, and to acquit otherwise. The question is what "your information" is.

A sincere voter treats her information as just her signal. A strategic voter does something different: she conditions on the event that her vote is pivotal — that is, the event that her vote actually changes the outcome.

Under unanimity, the defendant is convicted only if all three vote to convict. You are pivotal if and only if the other two jurors have voted to convict — which, since they are sincere, means they received signals $g$ and $g$.

Consider the four possible signal profiles for the other two jurors: $ii$, $ig$, $gi$, and $gg$. In the first three cases, at least one of the other jurors votes to acquit, so the defendant is acquitted regardless of your vote. Your vote is irrelevant. Only in the case where the other two both received $g$ does your vote determine the outcome: if you vote to convict, the defendant is convicted; if you vote to acquit, the defendant is acquitted.

A strategic voter therefore asks: given that I received signal $i$ and the other two jurors received $g$ and $g$, what is the probability the defendant is guilty?

The Calculation

The strategic voter's information is the signal profile $\{i, g, g\}$ — her own signal and the signals she infers from the pivotality event. This is different from the sincere voter's information, which was just $\{i, \cdot, \cdot\}$.

Using Bayes' rule with equal priors:

$\text{Prob}(G \mid igg) = \frac{\tfrac{1}{2} \cdot p^2(1-p)}{\tfrac{1}{2} \cdot p^2(1-p) + \tfrac{1}{2} \cdot p(1-p)^2}$

The numerator is the joint probability of observing signals $i$, $g$, $g$ when the state is $G$: the strategic voter gets $i$ (probability $1-p$), the first sincere juror gets $g$ (probability $p$), and the second gets $g$ (probability $p$), giving $p^2(1-p)$, times the prior $\frac{1}{2}$.

The denominator adds the corresponding term for state $I$: the strategic voter gets $i$ (probability $p$ when state is $I$), the other two get $g$ (probability $1-p$ each), giving $p(1-p)^2$, times $\frac{1}{2}$.

Dividing numerator and denominator by $\frac{1}{2} p(1-p)$:

$\text{Prob}(G \mid igg) = \frac{p}{p + (1-p)} = p$

The probability that the defendant is guilty, conditional on the strategic voter receiving signal $i$ and being pivotal, is exactly $p$.

Since $p \geq q$ (by the threshold assumption $1-p < q \leq p$), the strategic voter votes to convict — even though she received the innocent signal $i$.

This is the key result. A sincere juror with signal $i$ would compute $\text{Prob}(G \mid i) = 1-p < q$ and vote to acquit. A strategic juror with the same signal, conditioning on being pivotal, computes $\text{Prob}(G \mid igg) = p \geq q$ and votes to convict.

The Winner's Curse and the Swing Voter's Curse

This result has a striking parallel in auction theory.

In a common-value auction, the object's true value is unknown but the same for all bidders. Each bidder receives a noisy private signal about the value and bids accordingly. The highest bidder wins. But winning is bad news: to win, you must have bid higher than everyone else, which means your signal was higher than everyone else's — and since the true value is the average of all signals, your signal was probably an overestimate. A rational bidder should bid as if she has just learned that her signal is the highest of all bidders — and shade her bid downward accordingly. Winning an auction is a signal that you may have overpaid. This is the winner's curse.

Under unanimity, being pivotal is exactly analogous. To be pivotal — to be the juror whose vote determines the outcome — you must be in the situation where all other jurors voted to convict. Under unanimity, all others voting to convict means all others received signal $g$. If you received signal $i$ and are pivotal, you are the only person in the room who received an innocent signal. The accumulated evidence from the other two jurors overwhelmingly points to guilt. Your own $i$ signal is almost certainly noise.

This is the swing voter's curse: under unanimity, the event of being pivotal is itself evidence that your private signal is wrong. A rational strategic juror should therefore disregard her $i$ signal and vote to convict — not because she is corrupt, but because the pivotality event is more informative than her own signal.

Swing Voter's Curse: Under unanimity, being pivotal reveals that your $i$ signal is likely wrong. A rational strategic juror ignores it and votes to convict.

Sincere Voting Was Not Without Loss of Generality — Under Unanimity

The contrast with simple majority deserves emphasis. Under simple majority, a strategic juror who conditions on being pivotal still wants to vote sincerely. The pivotality event under simple majority (exactly one other juror votes to convict, and yours is the decisive vote) does not systematically shift the posterior far from what the signal alone would imply. The sincere-voting analysis was therefore without loss of generality for simple majority.

Under unanimity, the pivotality event is so extreme — requiring all others to have voted to convict — that it completely dominates the juror's own signal. A sincere $i$ vote is no longer rational.

This shows that the rationality of sincere voting is a property of the voting rule, not just of the voters. Simple majority is "sincere-compatible" in a way that unanimity is not. The institutional design shapes not just the outcomes but the behavior it makes rational.