Information Aggregation: Bayesian Updating

Information Aggregation: Bayesian Updating

Does One Vote Matter?

Before we can ask whether democracy aggregates information well, we should confront a more basic question: does an individual vote matter at all?

Consider a simple model: there are $N$ voters, each of whom votes for candidate $D$ or $R$ independently with probability $\frac{1}{2}$. The winner is determined by majority rule. Your vote is pivotal — it actually changes the outcome — only when the other $N-1$ voters are exactly tied.

For $N = 3$: you and two other voters. The other two voters each vote D or R with probability $\frac{1}{2}$. There are four equally likely outcomes for their votes: (D,D), (D,R), (R,D), (R,R). Your vote is pivotal in exactly two of them — (D,R) and (R,D), where the others tie. So:

$\text{Prob}(\text{pivotal}) = \frac{2}{4} = \frac{1}{2}$

For $N = 5$: the other four voters must tie (2 for D, 2 for R). The number of ways this happens is $\binom{4}{2} = 6$, out of $2^4 = 16$ total equally likely profiles:

$\text{Prob}(\text{pivotal}) = \frac{6}{16} = \frac{3}{8}$

This is less than $\frac{1}{2}$. As $N$ grows, the probability of a tie among the other voters falls rapidly. For $N = 1{,}000$, the pivot probability is approximately $2.5\%$. For $N = 10{,}000{,}000$, it is approximately $0.025\%$. In large electorates, any single vote is vanishingly unlikely to be decisive.

Yet democracies function. People vote — often in large numbers — and collective decisions seem to track the will of the electorate. If individual votes are so rarely decisive, why does democracy work at all?

One compelling answer is that the question is wrong. The value of voting is not about whether your single vote changes the outcome — it probably will not. The value is that when many people each cast an informed vote, the aggregate of those votes encodes vast amounts of dispersed information about the relative merits of the alternatives. Democracy works not because each vote matters individually, but because votes collectively aggregate information.

This motivates the next several lectures. We will build a formal model of information aggregation through voting and ask: can a group of voters, each with noisy private information, make a correct collective decision? We begin with the tools needed to answer that question.

The Framework: States, Priors, Signals, and Posteriors

To model information aggregation, we need a framework for representing uncertainty and learning.

State of the world: Something is true but unknown. We represent this as a random variable $\theta$ taking values in some set. In our main application, the state is binary: either the defendant is Guilty ($G$) or Innocent ($I$). In later applications to elections, the state might be which of two policies ($A$ or $B$) is better for society.

Prior belief: Before receiving any information, a decision-maker has a prior — a probability distribution over the possible states. We write $\text{Prob}(G)$ for the prior probability of guilt. In our model, we assume equal priors: $\text{Prob}(G) = \text{Prob}(I) = \frac{1}{2}$.

Signal: A signal is a piece of new information, potentially informative about the state. Signals are imperfect: they correlate with the true state but do not reveal it perfectly. In the juror model, each juror receives a binary signal $s \in \{g, i\}$: the signal $g$ suggests guilt, the signal $i$ suggests innocence.

Posterior belief: After receiving a signal, the decision-maker updates her prior using Bayes' rule to form a posterior — a revised probability distribution over states given the signal. The posterior is the correct probability of each state given all available information.

This framework — prior, signal, posterior — describes rational learning from new information. It is one of the most useful structures in all of economics and statistics.

The Juror Model

We will use the following model throughout Lectures 10–14. A court must decide whether to convict or acquit a defendant. The true state is either $G$ (guilty) or $I$ (innocent), each equally likely: $\text{Prob}(G) = \text{Prob}(I) = \frac{1}{2}$.

Each juror receives a private signal $s \in \{g, i\}$ about the defendant's guilt. The signal is binary and symmetric with accuracy $p > \frac{1}{2}$:

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

In words: if the defendant is guilty, each juror's signal points toward guilt with probability $p$ and toward innocence with probability $1-p$. If the defendant is innocent, each juror's signal points toward innocence with probability $p$ and toward guilt with probability $1-p$. The signal is accurate more often than not, since $p > \frac{1}{2}$.

The signals of different jurors are independent conditional on the state — knowing one juror's signal tells you nothing about another juror's signal beyond what you already knew about the state. This is the key statistical assumption that allows information to aggregate.

Bayesian Updating

Suppose a juror receives signal $g$ (suggesting guilt). How should she update her belief about the defendant's guilt? She starts with prior $\text{Prob}(G) = \frac{1}{2}$ and needs to compute $\text{Prob}(G \mid g)$.

Applying Bayes' rule:

$\text{Prob}(G \mid g) = \frac{\text{Prob}(g \mid G) \cdot \text{Prob}(G)}{\text{Prob}(g \mid G) \cdot \text{Prob}(G) + \text{Prob}(g \mid I) \cdot \text{Prob}(I)}$

Substituting the numbers:

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

So after seeing signal $g$, the juror's posterior belief that the defendant is guilty is exactly $p$.

Now suppose the juror receives signal $i$ (suggesting innocence). By the same calculation:

$\text{Prob}(G \mid i) = \frac{\text{Prob}(i \mid G) \cdot \text{Prob}(G)}{\text{Prob}(i \mid G) \cdot \text{Prob}(G) + \text{Prob}(i \mid I) \cdot \text{Prob}(I)} = \frac{(1-p) \cdot \frac{1}{2}}{(1-p) \cdot \frac{1}{2} + p \cdot \frac{1}{2}} = 1-p$

After seeing signal $i$, the posterior probability of guilt is $1-p < \frac{1}{2}$.

These two results — $\text{Prob}(G \mid g) = p$ and $\text{Prob}(G \mid i) = 1-p$ — are the key outputs of Bayesian updating in this model. The signal $g$ raises the juror's posterior above $\frac{1}{2}$, while the signal $i$ lowers it below $\frac{1}{2}$. Importantly, the size of the shift is governed by $p$: a more accurate signal (higher $p$) shifts the posterior more.

Bayes' Rule in General

The general form of Bayes' rule is worth stating clearly. For any two events $A$ and $B$:

$\text{Prob}(A \mid B) = \frac{\text{Prob}(B \mid A) \cdot \text{Prob}(A)}{\text{Prob}(B \mid A) \cdot \text{Prob}(A) + \text{Prob}(B \mid A^c) \cdot \text{Prob}(A^c)}$

Here $A^c$ denotes the complement of $A$. The denominator is simply the total probability of observing $B$, computed by the law of total probability. Bayes' rule inverts a conditional probability: it tells you $\text{Prob}(A \mid B)$ given that you know $\text{Prob}(B \mid A)$. In the juror model, you know how the signal is generated given the state (that is the signal accuracy $p$), and Bayes' rule tells you the probability of each state given the signal you observed.

Learning Always Helps

The Bayesian updating result has an immediate implication. Before receiving any signal, the juror's best assessment of guilt is the prior $\frac{1}{2}$. After receiving signal $g$, her assessment rises to $p > \frac{1}{2}$. After receiving signal $i$, it falls to $1-p < \frac{1}{2}$. In both cases, the juror's posterior is more extreme than $\frac{1}{2}$ and points in the direction of the signal.

In Lecture 11, we will show that a juror who acts on this updated belief — convicting when the posterior exceeds a threshold and acquitting otherwise — makes better decisions than a juror who ignores the signal entirely. The signal is always useful: it shifts the posterior in the right direction on average. Even an imperfect signal with $p$ slightly above $\frac{1}{2}$ improves decision quality compared to no information at all.

This is a general principle of Bayesian reasoning: more information never hurts a rational decision-maker. Any additional signal, no matter how noisy, weakly improves expected decision quality.

The key question for the rest of this module is: what happens when we aggregate the signals of many jurors? Can a group do better than an individual? And does the choice of voting rule matter for the quality of the collective decision? These are the questions of Lectures 11 through 14.