Political Persuasion: Model Setup

Political Persuasion: Model Setup

A Different Kind of Information Problem

The lectures on juries and elections studied a world in which information arrives from nature. Each juror independently draws a private signal — a noisy but unmanipulated observation correlated with the true state of the world. The question was how to aggregate these signals into a correct collective decision. The information channel was passive: nature drew the signals, and voters simply received them.

We now turn to a fundamentally different environment. The information channel is controlled by a strategic actor with his own agenda. A politician chooses what to reveal about himself. He is not trying to inform the voter — he is trying to win. The voter must therefore think not just about what was said, but why it was said, and crucially, what was left unsaid.

This shift in perspective — from passive signals to strategic communication — opens a new set of questions. How much can a strategic sender persuade a rational, skeptical receiver? What are the limits of persuasion when the receiver knows the sender is self-interested? And does the structure of the communication channel shape the answer?

Persuasion in Practice

These are not abstract questions. Strategic political communication is one of the most consequential features of democratic elections, and it has taken many forms across history.

In 1960, John F. Kennedy's campaign distributed approximately two million pamphlets on civil rights privately through African American churches in the South and urban North. The pamphlet — known as the "blue bomb" — was not broadcast publicly; it was targeted specifically at a constituency whose turnout and support would be decisive. The message was tailored to its audience and deliberately withheld from audiences who might have reacted negatively.

In 2004, the Bush re-election campaign used direct mail to send traditional family values messaging specifically to evangelical Christians. Different voters received different communications, each calibrated to the concerns and values of that particular group.

In 2016, the political data firm Cambridge Analytica claimed to have built psychological profiles of tens of millions of Americans and used them to deliver micro-targeted advertisements for the Brexit referendum and the Trump campaign. The same candidate could be presented differently to voters identified as having different personality types, anxieties, and political preoccupations.

These examples share a common structure: one party with private information (the candidate, knowing his true policy positions, background, and intentions) communicates strategically with another party who lacks that information (the voter). The communication is designed not to inform but to persuade.

Today's lecture focuses on the simplest version of this problem: one challenger and one voter. This is the entry point into the model. The extension to multiple voters with targeted advertising comes next.

The Model

Players and Timeline

There are two strategic players and one passive benchmark:

The Challenger knows his policy position $x \in [-1, 1]$. This policy represents where he stands on a left-right spectrum. He sends a message to the voter. His sole objective is to win — he receives a payoff of 1 if the voter approves him and 0 if she rejects him.

The Voter does not know the challenger's policy $x$. She observes only the challenger's message and uses it to decide whether to approve or reject him. If she rejects, the incumbent wins by default.

The Incumbent is not a strategic player. He represents the status quo, with a policy position of $0$ — the center of the policy space.

The timeline is sequential:

1. The challenger learns his policy $x$.
2. The challenger sends a message to the voter.
3. The voter observes the message and decides to approve or reject.

The challenger's policy $x$ is the state of the world from the voter's point of view: it is what she cares about, it is unknown to her, and it determines what she would receive if she approved him.

Prior Information

The voter has no prior knowledge about the challenger's specific policy. She believes, before receiving any message, that $x$ is drawn from the uniform distribution on $[-1, 1]$:

$x \sim U[-1, 1]$

This means she initially considers all policies equally likely. The policy space $[-1, 1]$ is symmetric around the status quo $0$, with $-1$ representing the extreme left and $1$ the extreme right.

The Voter's Preferences

The voter has a bliss point $v \in [-1, 1]$ — the policy she would most prefer. The further a policy is from $v$, the less she likes it. Her payoffs reflect this in a natural way:

$u_v(\text{approve},\, x) = -|v - x|, \qquad u_v(\text{reject}) = -|v|$

Approving a challenger with policy $x$ gives her payoff $-|v - x|$: the negative of the distance between her bliss point and the challenger's policy. Rejecting gives her payoff $-|v|$: the negative of the distance between her bliss point and the status quo policy $0$, since rejection means the incumbent (with policy $0$) wins.

The net payoff from approval — the gain from approving relative to rejecting — is:

$u_v(\text{approve}, x) - u_v(\text{reject}) = -|v - x| - (-|v|) = |v| - |v - x|$

The voter approves the challenger if and only if this net payoff is non-negative:

$|v| - |v - x| \geq 0 \iff |v - x| \leq |v|$

In words: the voter approves the challenger if the challenger's policy is no further from her bliss point than the status quo is. The status quo policy of $0$ is the benchmark she compares against.

A few observations are immediate from the net payoff expression:

• If $v < 0$, the voter is a left-leaning voter: her bliss point is to the left of the status quo.
• If $v > 0$, she is a right-leaning voter.
• If $v = 0$, her bliss point is the status quo itself — she is indifferent between approving and rejecting any challenger whose policy is $0$, and prefers rejection to any challenger with a non-zero policy.
• The further $|v|$ is from zero, the more extreme the voter, and the wider the set of challenger policies she would approve.

The Net Payoff as a Function of Challenger Policy

To see the shape of the voter's decision problem, it helps to graph the net payoff from approval as a function of $x$ for a specific bliss point.

Take a moderately right-leaning voter with $v = \frac{1}{4}$. Her net payoff from approving a challenger with policy $x$ is:

$f(x) = \left|\tfrac{1}{4}\right| - \left|\tfrac{1}{4} - x\right| = \tfrac{1}{4} - |x - \tfrac{1}{4}|$

This is a tent function, peaking at $x = v = \frac{1}{4}$ and decreasing linearly in both directions. Expanding:

$f(x) = \begin{cases} x & \text{if } x < \tfrac{1}{4} \\ \tfrac{1}{2} - x & \text{if } x \geq \tfrac{1}{4} \end{cases}$

Key values: $f(0) = 0$ (indifferent at status quo), $f\!\left(\tfrac{1}{4}\right) = \tfrac{1}{4}$ (highest net payoff, at bliss point), $f\!\left(\tfrac{1}{2}\right) = 0$ (indifferent again), $f(1) = -\tfrac{1}{2}$ (strongly negative for the far right).

The voter approves any challenger whose policy $x$ falls in the interval $[0, \tfrac{1}{2}]$, and rejects any challenger outside this interval. The interval $[0, \tfrac{1}{2}]$ is symmetric around the voter's bliss point $v = \tfrac{1}{4}$: the acceptable range extends a distance of $\tfrac{1}{4}$ in each direction from $v$, which is exactly the distance from $v$ to the status quo.

This tent function structure is general: for any bliss point $v > 0$, the voter approves challengers with policies in $[0, 2v]$ and rejects all others. The approval region is determined by the voter's distance from the status quo, not just by her bliss point in isolation.

Communication

The challenger knows his policy $x$ and must decide what to communicate. A message $m$ is a subset of $[-1, 1]$ that contains $x$:

$m \subseteq [-1, 1], \quad x \in m$

This constraint encodes a crucial assumption: the challenger cannot lie by commission. He cannot claim to have a policy he does not have. If his policy is $x = 0.3$, he cannot send the message $[-0.5, 0]$, because $0.3 \notin [-0.5, 0]$.

What he can do is be vague. He can send a wider interval that includes $x$ along with many other possible policies. The voter, upon receiving the message, knows only that the challenger's policy is somewhere in $m$ — she cannot pinpoint exactly where.

Some examples illustrate the range of possible messages:

• $m = [-\tfrac{1}{2},\, 0]$: "I am moderately left." The challenger reveals that he is somewhere in the left half of the moderate range.
• $m = [-\tfrac{1}{4},\, \tfrac{1}{4}]$: "I am a moderate." The challenger claims to be near the center.
• $m = [-1,\, 1]$: The fully uninformative message — "I have a policy." This tells the voter nothing beyond the fact that the challenger exists, since every possible policy is in $[-1, 1]$ by definition.

The key asymmetry: the challenger can lie by omission (by being vague) but not by commission (by misrepresenting his location). He can hide inside a wide interval but cannot claim to be somewhere he is not.

The Limits to Persuasion

The central question of this model is: how does a rational, self-interested challenger maximize his probability of approval?

This is a question about the limits to persuasion — how much can strategic communication achieve when the receiver is rational and the sender cannot lie outright?

Before answering this question, it is worth noting how broadly the model's assumptions apply. The structure — an informed sender who can send messages at will, a receiver who chooses between two actions, and a sender whose payoff depends only on the receiver's action — describes many settings beyond electoral politics:

• A job applicant communicates with a potential employer. The applicant knows his qualifications; the employer does not. The applicant wants the job offer (one action) and will not get it if the employer learns he is underqualified (the other action).
• A country seeking to deter an adversary from entering a conflict. The deterring country knows its own resolve; the adversary does not. The deterring country wants the adversary to stand down.
• A seller communicating the quality of a product to a buyer who cannot directly observe it.

In all of these cases, the same logic applies: the sender has private information, cannot lie outright (or faces reputational costs from being caught), and wants to influence the receiver's binary decision. The limits to persuasion are the same in each.

The next lecture works through the challenger's optimal strategy — what message to send, and why strategic vagueness turns out to be the answer.