From Juries to Elections: Voting Rules and Error Tradeoffs

From Juries to Elections: Voting Rules and Error Tradeoffs

From the Courtroom to the Ballot Box

The jury model we have been developing might seem narrowly applicable — a stylized account of criminal trials, not a general theory of democratic voting. But the framework is more broadly applicable than it first appears, and this lecture makes that connection explicit.

In the jury model, the key features are: a state of the world that is true but unknown; voters (jurors) who each receive noisy signals about the true state; and a social choice function that aggregates individual votes into a collective decision. The question is whether the aggregation process produces the right answer.

Now consider an election with a common-interest structure. Two policies, $A$ and $B$, are under consideration, and one of them is genuinely better for society — but which one is better is uncertain. Citizens have private information (their experiences, observations, expertise) about which policy is superior. They vote, and the majority determines policy. Does the vote aggregate their dispersed information correctly?

The translation between the two settings is:

• Jurors $\to$ voters
• State $G/I$ $\to$ state $A/B$ (which policy is better)
• Signal $g/i$ $\to$ noisy private information about $A$ vs. $B$
• Convict/acquit $\to$ vote for $A$ or $B$
• Social choice function $\to$ electoral rule

This translation works whenever voters share a common interest — they all want the better policy to be chosen, and they disagree only because they have different information, not different values. Not every election has this structure: elections about redistribution or immigration involve fundamental value disagreements, not just information differences. But many elections do involve a substantial common-interest component — which candidate is more competent, which economic policy will be more effective, which public health intervention is better — and in these settings the jury model applies directly.

Simple Majority: The Optimal Rule for Minimizing Total Error

We have already computed the simple majority performance. But there is a deeper result we can now state:

Simple majority minimizes the overall probability of mistake across all aggregation rules (given fixed $N$, $p$, and symmetric priors).

We state this without full proof, but the intuition is clear: simple majority gives equal weight to each juror's signal, and since all signals are equally informative and independent, equal weighting is optimal. Any departure from equal weighting — giving some votes more influence than others, or requiring more than a majority for conviction — either wastes information (by overweighting some signals) or systematically biases the outcome (by requiring more evidence for one type of decision than the other).

Simple majority is the neutral rule — it treats the two types of error symmetrically and minimizes the total probability of error. But as we saw in Lecture 13, symmetry is not always what we want.

Vasili Arkhipov and the Value of Unanimity

On October 27, 1962, at the height of the Cuban Missile Crisis, a Soviet submarine designated B-59 was operating near Cuba. It had been out of contact with Moscow for days and was taking depth charges from US destroyers attempting to force it to surface. The crew did not know whether World War III had begun.

The submarine's captain, Valentin Savitsky, concluded that war had started and ordered the launch of a nuclear torpedo. Under Soviet naval protocol, the launch required the agreement of three senior officers on board. Two of them agreed. The third was Vasili Arkhipov, second-in-command of the entire submarine flotilla. Arkhipov refused to authorize the launch. Without his agreement, the torpedo was never fired.

The Soviet Union and the United States did not go to nuclear war. Arkhipov is sometimes called the man who saved the world.

This episode illustrates the logic of unanimity rules with devastating clarity. The decision being contemplated — launching a nuclear weapon — was potentially catastrophic and irreversible. The potential mistake to be avoided was a false positive: launching when war had not actually begun. The two officers who agreed had received misleading signals (depth charges that they interpreted as acts of war; in fact they were warning signals). Arkhipov had a different assessment. The unanimity protocol, requiring all three to agree, prevented the catastrophic error.

This is exactly the logic of unanimity in our model: it sets a very high bar for the "convict" decision (launch), making Type I errors (false positives) extremely rare, at the cost of making Type II errors (failing to launch when it would have been warranted) more common. When the costs of the two error types are radically asymmetric — when one type of mistake is catastrophic and irreversible, while the other is merely costly and correctable — unanimity or supermajority rules are rational design choices.

Unanimity Rule: The Full Derivation for $N=3$

Let us now work through the unanimity calculation systematically, as we did for simple majority in Lecture 13.

Under unanimity, the jury convicts only when all three jurors vote to convict — that is, only when all three signals are $g$.

When $\theta = G$: each juror independently gets $g$ with probability $p$. All three getting $g$ has probability $p^3$.

$\text{Prob}(\text{convict} \mid G) = p^3$

When $\theta = I$: each juror independently gets $g$ with probability $1-p$. All three getting $g$ has probability $(1-p)^3$.

$\text{Prob}(\text{convict} \mid I) = (1-p)^3$

The complementary probabilities:

$\text{Prob}(\text{acquit} \mid G) = 1-p^3 \qquad \text{(Type II error)}$

$\text{Prob}(\text{acquit} \mid I) = 1-(1-p)^3 \qquad \text{(correct acquittal)}$

Overall accuracy with equal priors:

$\text{Prob}(\text{correct}) = \frac{1}{2} \cdot p^3 + \frac{1}{2} \cdot \bigl[1-(1-p)^3\bigr]$

Overall mistake probability:

$\text{Prob}(\text{mistake}) = \frac{1}{2}(1-p)^3 + \frac{1}{2}(1-p^3)$

Side-by-Side Comparison at $p = 0.6$, $N = 3$

Here are the six key quantities for both rules when $p = 0.6$:

Simple majority:

• $\text{Prob}(\text{convict} \mid G) = 3(0.6)^2 - 2(0.6)^3 = 1.08 - 0.432 = 0.648$
• $\text{Prob}(\text{acquit} \mid I) = 0.648$
• $\text{Prob}(\text{convict} \mid I) = 0.352$ (Type I error)
• $\text{Prob}(\text{acquit} \mid G) = 0.352$ (Type II error)
• $\text{Prob}(\text{correct}) = 0.648$
• $\text{Prob}(\text{mistake}) = 0.352$

Unanimity:

• $\text{Prob}(\text{convict} \mid G) = (0.6)^3 = 0.216$
• $\text{Prob}(\text{acquit} \mid I) = 1-(0.4)^3 = 0.936$
• $\text{Prob}(\text{convict} \mid I) = (0.4)^3 = 0.064$ (Type I error)
• $\text{Prob}(\text{acquit} \mid G) = 1-(0.6)^3 = 0.784$ (Type II error)
• $\text{Prob}(\text{correct}) = \frac{1}{2}(0.216) + \frac{1}{2}(0.936) = 0.576$
• $\text{Prob}(\text{mistake}) = 0.424$

The contrast is stark. Unanimity reduces the Type I error rate from $35.2\%$ to $6.4\%$ — a fivefold reduction in the wrongful conviction rate. This comes at the cost of a massive increase in the Type II error rate, from $35.2\%$ to $78.4\%$. Under unanimity, more than three-quarters of guilty defendants will go free.

Overall accuracy falls from $64.8\%$ to $57.6\%$: unanimity is less accurate in total. But if wrongful conviction is considered roughly five times worse than wrongful acquittal, unanimity would be the preferred rule on expected-cost grounds.

Priorities and Real-World Institutions

Different societies and different decision contexts have different priorities. Real-world voting rules reflect implicit or explicit judgments about these priorities.

When minimizing Type I error is the priority: the appropriate standard is unanimity or a supermajority. Real-world examples include:

The US criminal jury requires a unanimous verdict of 12 jurors to convict. The legal standard is "beyond a reasonable doubt" — a very high threshold, reflecting the principle that the state's power to imprison citizens demands the strongest possible evidence.

The UN Security Council requires unanimity among the five permanent members (the P5 veto). A single permanent member can block any resolution, however strongly supported by other members. This prevents great powers from being coerced into military action — an irreversible, high-stakes decision — by a majority.

The European Council requires consensus of all 27 member states for many fundamental EU decisions. Expanding the union, changing foundational treaties, admitting new members — these are irreversible decisions where a single vetoing state can block action. The rule reflects the asymmetric costs of forcing a state into a union it does not consent to.

Amending the US Constitution requires two-thirds of both houses of Congress plus three-quarters of the states — an extraordinarily high bar that makes constitutional change nearly impossible without broad consensus. The Framers designed this threshold deliberately to prevent impulsive or factional amendments.

William Blackstone wrote in 1769: "Better that ten guilty persons escape than that one innocent suffer." This is not a mathematical claim about accuracy — it is a statement of values. It says that the disutility of convicting an innocent person is at least ten times greater than the disutility of acquitting a guilty one. Under this asymmetric loss function, unanimity (or a high standard of proof) is rational.

When treating both errors symmetrically is the priority: the appropriate rule is simple majority. Real-world examples include ordinary legislative votes (US House, Senate, Supreme Court, UK House of Commons), most referenda and elections, corporate shareholder votes, and many administrative decisions. In these settings, the two possible mistakes — failing to pass a good law, or passing a bad law — are considered roughly equally costly and reversible.

The US standard of proof in civil cases is "preponderance of the evidence" — meaning probability greater than $\frac{1}{2}$. This corresponds almost exactly to the simple majority rule in our model: the plaintiff wins if the evidence tips the balance toward their account, even slightly.

The Central Lesson: Values, Not Math

The comparison between simple majority and unanimity does not have an objectively correct resolution. The right voting rule depends entirely on how a society values the two types of error — and that is a question of values, not mathematics.

Mathematics can tell us exactly what each rule achieves: the Type I error rate, the Type II error rate, the overall accuracy. Mathematics can tell us how these quantities change with jury size and signal accuracy. Mathematics can identify which rule minimizes total error (simple majority) and which minimizes Type I error for a given overall accuracy (unanimity/supermajority).

But mathematics cannot tell us which error is worse. That depends on moral commitments: how much we value protecting the innocent versus how much we value punishing the guilty; how much weight we give to individual rights versus collective efficiency; how we weigh irreversible harms against recoverable ones.

The design of democratic institutions — how many votes are needed to pass a law, to convict a criminal, to override a presidential veto, to amend a constitution — is at its core a question about these values. The tools developed in this module can make the tradeoffs precise and the consequences of different choices clear. But the choice itself belongs to the political community, not to the theorist.

This is the closing theme of the course. Formal models do not replace political judgment — they discipline it. They force us to be clear about what we assume, explicit about what follows from those assumptions, and honest about what remains beyond the model's reach.