Electoral Competition: Uncertainty and Polarization

Electoral Competition: Uncertainty and Polarization

Why Uncertainty Matters

Models 1 and 2 both predict that candidates converge to the median voter's bliss point $v^M$. This prediction fails dramatically in practice — democratic elections regularly feature substantial ideological polarization between the major parties. We have modified candidates' motivations (from office to policy) without breaking convergence. What else could drive divergence?

The key assumption we have not yet relaxed is that candidates know the location of $v^M$ with certainty. In reality, no candidate knows exactly where the median voter stands. Polling is imprecise, the electorate shifts between elections, and turnout is uncertain. Candidates face genuine uncertainty about the preferences of the electorate they are trying to win.

The third model incorporates this uncertainty and shows that it can rationalize equilibrium policy divergence. Crucially, the degree of divergence is inversely related to the degree of uncertainty — when uncertainty is high, candidates diverge; when uncertainty is low, candidates converge.

Model 3: Uncertain Median Voter

Setup

There are two candidates, $D$ and $R$, with bliss points $z_D = -1$ and $z_R = 1$ at the extremes of the policy space $[-1, 1]$. Each chooses a platform $x_i \in [-1, 1]$. Payoffs are policy-motivated, as in Model 2.

The key change: the median voter's bliss point $v^M$ is unknown. Candidates share a common prior belief that $v^M$ is drawn uniformly from the interval $\left[-\frac{1}{2b}, \frac{1}{2b}\right]$, where $b \in \left[\frac{1}{2}, +\infty\right)$ is a parameter measuring how concentrated beliefs about $v^M$ are.

When $b = \frac{1}{2}$, the interval is $[-1, 1]$ — the entire policy space — and candidates have maximum uncertainty about $v^M$. When $b = 1$, the interval is $[-\frac{1}{2}, \frac{1}{2}]$ — moderate uncertainty. As $b \to +\infty$, the interval collapses to the single point $\{0\}$, and candidates know $v^M = 0$ with certainty. This limiting case recovers Model 2 (with $v^M = 0$).

The parameter $b$ can be thought of as measuring the precision of candidate polling and intelligence about voter preferences.

The Probability That D Wins

Given platforms $(x_D, x_R)$, candidate $D$ wins iff the median voter prefers $D$'s platform. Since preferences are symmetric, this occurs when $v^M < \frac{x_D + x_R}{2}$ — the median voter is to the left of the midpoint of the two platforms.

Since $v^M \sim \text{Uniform}\left[-\frac{1}{2b}, \frac{1}{2b}\right]$, we can compute the probability $P_D$ that $D$ wins:

• If the midpoint $\frac{x_D + x_R}{2}$ is below the entire support — i.e., $\frac{x_D + x_R}{2} < -\frac{1}{2b}$ — then $v^M$ is always above the midpoint, $D$ never wins: $P_D = 0$.
• If the midpoint is above the entire support — i.e., $\frac{x_D + x_R}{2} > \frac{1}{2b}$ — then $v^M$ is always below the midpoint, $D$ always wins: $P_D = 1$.
• If the midpoint lies inside the support — the interesting case — then $P_D$ equals the fraction of the uniform distribution lying to the left of the midpoint:

$P_D = \frac{\frac{x_D + x_R}{2} - \left(-\frac{1}{2b}\right)}{\frac{1}{2b} - \left(-\frac{1}{2b}\right)} = \frac{\frac{x_D + x_R}{2} + \frac{1}{2b}}{\frac{1}{b}} = \frac{b(x_D + x_R) + 1}{2}$

Collecting all three cases:

$P_D(x_D, x_R) = \begin{cases} 0 & \text{if } \dfrac{x_D+x_R}{2} < -\dfrac{1}{2b} \\[8pt] \dfrac{b(x_D+x_R)+1}{2} & \text{if } \dfrac{x_D+x_R}{2} \in \left[-\dfrac{1}{2b},\, \dfrac{1}{2b}\right] \\[8pt] 1 & \text{if } \dfrac{x_D+x_R}{2} > \dfrac{1}{2b} \end{cases}$

We will focus on the interior (second) case, which is where the interesting equilibrium behavior occurs.

D's Expected Payoff

With policy-motivated payoffs and $z_D = -1$, $D$'s expected utility is:

$EU_D(x_D, x_R) = P_D \cdot (-|x_D - z_D|) + (1 - P_D) \cdot (-|x_R - z_D|)$

For $x_D \in [-1, 1]$ and $z_D = -1$, we have $x_D \geq z_D = -1$, so $|x_D - z_D| = x_D - (-1) = x_D + 1$. Similarly, $|x_R - z_D| = x_R + 1$.

Substituting the interior formula for $P_D$:

$EU_D = -\frac{b(x_D+x_R)+1}{2}(x_D + 1) - \left(1 - \frac{b(x_D+x_R)+1}{2}\right)(x_R + 1)$

Let us expand this carefully. Write $P = \frac{b(x_D+x_R)+1}{2}$ for brevity. Then:

$EU_D = -P(x_D+1) - (1-P)(x_R+1) = -P(x_D+1) - (x_R+1) + P(x_R+1)$ $= P\left[(x_R+1) - (x_D+1)\right] - (x_R+1) = P(x_R - x_D) - (x_R+1)$

Substituting $P = \frac{b(x_D+x_R)+1}{2}$:

$EU_D = \frac{b(x_D+x_R)+1}{2}(x_R - x_D) - (x_R+1)$

Expanding:

$EU_D = \frac{b(x_D+x_R)(x_R-x_D)}{2} + \frac{x_R - x_D}{2} - x_R - 1$

$= \frac{b(x_R^2 - x_D^2)}{2} + \frac{x_R - x_D}{2} - x_R - 1$

$= -\frac{b}{2}x_D^2 - \frac{1}{2}x_D + \frac{b}{2}x_R^2 - \frac{1}{2}x_R - 1$

The terms involving only $x_R$ are constants from $D$'s perspective. Writing out only the terms that depend on $x_D$:

$EU_D = -\frac{b}{2}x_D^2 - \frac{1}{2}x_D + \text{(terms in } x_R \text{ only)}$

D's Best Response

To maximize $EU_D$ over $x_D$, take the derivative with respect to $x_D$ and set it to zero:

$\frac{\partial EU_D}{\partial x_D} = -bx_D - \frac{1}{2} = 0$

Solving:

$x_D^* = -\frac{1}{2b}$

This is a maximum because the second derivative $\frac{\partial^2 EU_D}{\partial x_D^2} = -b < 0$. Remarkably, $x_D^*$ does not depend on $x_R$: $D$'s best response is the same regardless of where $R$ chooses to position. In game theory parlance, $x_D^* = -\frac{1}{2b}$ is a dominant strategy for $D$ (in the interior region).

R's Best Response

By symmetry, swapping $D \leftrightarrow R$ and $z_D \leftrightarrow z_R$ (with $z_R = 1$), $R$'s expected payoff in the interior is:

$EU_R = -\frac{b}{2}x_R^2 + \frac{1}{2}x_R + \text{(terms in } x_D \text{ only)}$

Differentiating and setting to zero:

$\frac{\partial EU_R}{\partial x_R} = -bx_R + \frac{1}{2} = 0 \implies x_R^* = \frac{1}{2b}$

Again, $R$'s best response does not depend on $x_D$.

The Unique Nash Equilibrium

Since both best responses are independent of the opponent's action, the Nash equilibrium is uniquely:

$x_D^* = -\frac{1}{2b}, \qquad x_R^* = \frac{1}{2b}$

We should verify that this equilibrium falls inside the interior region: the midpoint is $\frac{x_D^* + x_R^*}{2} = 0 \in \left[-\frac{1}{2b}, \frac{1}{2b}\right]$. Yes — the equilibrium midpoint is at the center of the support, so the interior-case formula applies and the equilibrium is valid.

Polarization and the Uncertainty Parameter

The equilibrium degree of polarization is:

$x_R^* - x_D^* = \frac{1}{2b} - \left(-\frac{1}{2b}\right) = \frac{1}{b}$

This formula shows how uncertainty drives polarization:

When $b = \frac{1}{2}$ (maximum uncertainty, $v^M$ uniform on $[-1,1]$): polarization is $\frac{1}{1/2} = 2$, which spans the entire policy space $[-1,1]$. Indeed, $x_D^* = -1 = z_D$ and $x_R^* = 1 = z_R$ — both candidates campaign on their own most preferred policies.

As $b$ increases (uncertainty decreases, the distribution of $v^M$ concentrates around 0): polarization $\frac{1}{b}$ decreases. Candidates move toward the center.

As $b \to \infty$ (certainty that $v^M = 0$): $x_D^* \to 0$ and $x_R^* \to 0$ — full convergence to the median voter. Model 3 recovers Model 2 as a special case.

The Economic Interpretation

Why does uncertainty generate divergence? When candidates know exactly where $v^M$ is, the electoral race is knife-edge: being even slightly further from $v^M$ than your opponent means losing for certain. This makes the electoral incentive enormous and forces both candidates all the way to $v^M$.

When $v^M$ is uncertain, the electoral outcome is a probability rather than a certainty. Moving slightly toward the center increases $D$'s win probability, but the gain is proportional to the density of $v^M$ near the midpoint, which is bounded (it is $b$ in the interior case). At the margin, $D$ balances the benefit of a higher win probability against the cost of implementing a worse policy if she wins. The optimal balance involves staying some distance from the center — closer when uncertainty is low (high $b$), further when uncertainty is high (low $b$).

This model provides a rational explanation for observed polarization: parties are not irrationally indulging extremist preferences, but rationally responding to uncertainty about voter preferences. Elections with better polling, more transparent public opinion, and more stable electorates should, all else equal, exhibit less polarization.