Juror's Optimal Action and Following the Signal

Juror's Optimal Action and Following the Signal

Recap: What We Know from Bayesian Updating

In Lecture 10 we derived the juror's posterior beliefs given her signal. Starting from a prior of $\text{Prob}(G) = \text{Prob}(I) = \frac{1}{2}$ and using a binary signal with accuracy $p > \frac{1}{2}$:

$\text{Prob}(G \mid g) = p \qquad \text{and} \qquad \text{Prob}(G \mid i) = 1-p$

The signal $g$ (pointing toward guilt) raises the posterior to $p$, which is above $\frac{1}{2}$. The signal $i$ (pointing toward innocence) lowers the posterior to $1-p$, which is below $\frac{1}{2}$.

Now we ask: given these updated beliefs, what should the juror do?

The Juror's Decision Problem

A juror must choose one of two actions: acquit or convict. She wants to make the correct decision — convict the guilty, acquit the innocent — but she cannot observe the true state directly, only through her signal.

To formalize the juror's preferences, we introduce a parameter $q \in (0,1)$ called the threshold of doubt. The juror convicts if and only if her posterior probability of guilt meets or exceeds this threshold:

$\text{Convict if } \text{Prob}(G) \geq q, \qquad \text{Acquit if } \text{Prob}(G) < q$

The threshold $q$ encodes the juror's (or society's) standard of proof — how confident she must be in the defendant's guilt before voting to convict. A very high threshold (say $q = 0.99$) requires near-certainty of guilt; a low threshold (say $q = 0.55$) requires only modest evidence.

Acting Without Learning

Before turning to how the juror uses her signal, let us consider the baseline: what does the juror do with only her prior belief?

Prior to receiving any signal, the juror's probability of guilt is $\frac{1}{2}$. Applying the threshold rule:

• If $q \leq \frac{1}{2}$: the prior satisfies the threshold, so the juror convicts without any signal. Every defendant gets convicted regardless of evidence.
• If $q > \frac{1}{2}$: the prior falls short of the threshold, so the juror acquits without any signal. Every defendant gets acquitted regardless of evidence.

In either case, the outcome is the same for all defendants, and the juror's decision is completely uninformative. This is the worst possible behavior — it ignores all available information.

Acting on the Signal

Now suppose the juror receives signal $g$ (posterior = $p$) or signal $i$ (posterior = $1-p$). She applies the threshold rule to each posterior:

After signal $g$: convict if $p \geq q$; acquit if $p < q$.

After signal $i$: convict if $1-p \geq q$; acquit if $1-p < q$.

The Condition for Following the Signal

The most informative and natural behavior is for the juror to follow her signal: convict after $g$ and acquit after $i$. This requires two conditions to hold simultaneously.

After signal $g$, convict: $p \geq q$.

After signal $i$, acquit: $1-p < q$.

Both conditions together can be written compactly as:

$1-p < q \leq p$

This is the following the signal condition. It says the threshold lies strictly above $1-p$ (so the juror acquits after seeing $i$) and weakly below $p$ (so the juror convicts after seeing $g$).

As a concrete example, take $p = 0.75$ and $q = 0.5$. Then $1-p = 0.25 < 0.5 = q \leq 0.75 = p$. The condition is satisfied. A juror with these parameters convicts after $g$ (posterior $0.75 \geq 0.5$) and acquits after $i$ (posterior $0.25 < 0.5$). She follows her signal.

We assume for the remainder of this lecture and all subsequent lectures that $1-p < q \leq p$. Under this assumption, the juror's action is fully determined by her signal: signal $g$ leads to conviction, signal $i$ leads to acquittal, with no exceptions.

This is both the most natural behavior — use your information — and the behavior that maximizes expected decision quality, as we now show.

The Six Quantities of Interest

Under the assumption that jurors follow their signals, we can compute six probabilities that characterize the quality of a juror's decisions. There are two types of correct decisions and two types of mistakes.

We organize the analysis around the four possible combinations of (true state, signal received):

State $I$, signal $i$: probability $\frac{1}{2} \cdot p$. The defendant is innocent and the juror gets the informative signal. She acquits — correct decision.

State $I$, signal $g$: probability $\frac{1}{2} \cdot (1-p)$. The defendant is innocent but the juror gets the misleading signal. She convicts — a mistake (wrongful conviction).

State $G$, signal $g$: probability $\frac{1}{2} \cdot p$. The defendant is guilty and the juror gets the informative signal. She convicts — correct decision.

State $G$, signal $i$: probability $\frac{1}{2} \cdot (1-p)$. The defendant is guilty but the juror gets the misleading signal. She acquits — a mistake (wrongful acquittal).

From these four joint probabilities, we derive the six key quantities:

Probability of conviction given guilt: This is the probability the juror gets signal $g$ when the defendant is actually guilty:

$\text{Prob}(\text{convict} \mid G) = p$

Probability of acquittal given innocence: This is the probability the juror gets signal $i$ when the defendant is actually innocent:

$\text{Prob}(\text{acquit} \mid I) = p$

These two quantities are equal — a consequence of the symmetric signal structure — and both equal $p$. This reflects the juror's ability to detect both truth and falsity.

Probability of a correct decision: The juror makes the correct decision when she convicts a guilty defendant or acquits an innocent one:

$\text{Prob}(\text{correct}) = \text{Prob}(G) \cdot \text{Prob}(\text{convict} \mid G) + \text{Prob}(I) \cdot \text{Prob}(\text{acquit} \mid I) = \frac{1}{2} \cdot p + \frac{1}{2} \cdot p = p$

Probability of conviction given innocence (wrongful conviction):

$\text{Prob}(\text{convict} \mid I) = 1-p$

Probability of acquittal given guilt (wrongful acquittal):

$\text{Prob}(\text{acquit} \mid G) = 1-p$

Probability of a mistake:

$\text{Prob}(\text{mistake}) = \frac{1}{2} \cdot (1-p) + \frac{1}{2} \cdot (1-p) = 1-p$

The Key Insight: Learning Always Beats Guessing

The results just derived have a stark and important interpretation. A juror who ignores her signal and simply guesses — say, always acquitting — would be correct exactly $\frac{1}{2}$ of the time (when the defendant is actually innocent) and wrong $\frac{1}{2}$ of the time. A juror who convicts without reason would similarly be correct $\frac{1}{2}$ of the time. Random guessing yields $\text{Prob}(\text{correct}) = \frac{1}{2}$.

A juror who follows her signal achieves $\text{Prob}(\text{correct}) = p > \frac{1}{2}$. She makes fewer mistakes: $\text{Prob}(\text{mistake}) = 1-p < \frac{1}{2}$.

The improvement over random guessing is entirely due to the signal. Even a very noisy signal — say, $p = 0.51$, barely better than a coin flip — improves decision quality above $\frac{1}{2}$. More accurate signals (higher $p$) lead to better decisions.

This is not a surprising result mathematically, but it carries a profound practical implication: gathering and using information always improves decision quality. A deliberation procedure that helps jurors form more accurate beliefs about guilt — better evidence presentation, clearer legal instructions, more time for reflection — will lead to fewer mistakes, regardless of the voting rule.

Preparing for Multiple Jurors

The analysis so far concerns a single juror acting on a single signal. But real juries have many members — twelve in a US criminal trial. The natural question is whether more jurors means better decisions. The answer, known as the Condorcet Jury Theorem, is the subject of Lecture 13.

Before we can state and prove that theorem, we need to extend the framework from a single juror to a panel of $N$ jurors. Each juror receives an independent signal with accuracy $p$, forms a posterior, and (under our assumption) follows her signal. The panel then aggregates the $N$ individual votes according to some social choice function — simple majority, unanimity, supermajority — to reach a verdict.

The key question is: how do different voting rules perform in terms of overall decision quality? And does a larger jury always perform better? We take up these questions in Lectures 12 and 13.