How a minority can win: Unrepresentative outcomes in a simple model of voter turnout

The outcome of an election depends not only on which candidate is more popular, but also on how many of their voters actually turn out to vote. Here we consider a simple model in which voters abstain from voting if they think their vote would not matter. Specifically, they do not vote if they feel sure their preferred candidate will win anyway (a condition we call complacency), or if they feel sure their candidate will lose anyway (a condition we call dejectedness). The voters reach these decisions based on a myopic assessment of their local network, which they take as a proxy for the entire electorate: voters know which candidate their neighbors prefer and they assume—perhaps incorrectly—that those neighbors will turn out to vote, so they themselves cast a vote if and only if it would produce a tie or a win for their preferred candidate in their local neighborhood. We explore various network structures and distributions of voter preferences and find that certain structures and parameter regimes favor unrepresentative outcomes where a minority faction wins, especially when the locally preferred candidate is not representative of the electorate as a whole.

Figure 1

The behavioral assumptions of (a) dejectedness, (b) complacency, and (c) their combination applied to the same ring network with a $5-4$ split between orange and purple nodes. (a) The top orange node is surrounded by a purple node on either side. Thus, in its local (one-hop) neighborhood it is outnumbered $2-1$, so its vote cannot tie or win the upcoming election in that local neighborhood. Making a myopic (and wrong) estimate of the orange opinion's chances globally, based solely on its local neighborhood, the orange node believes its vote cannot affect the upcoming election, so it gets dejected and does not cast a vote, as indicated by the gray cross. (b) The two bottom nodes are completely surrounded by other orange nodes. Based on this local information, they erroneously conclude that the upcoming election is a safe win, become complacent, and do not vote. (c) Because three orange nodes do not vote, purple wins the overall election by $4-2.$

Figure 2

Unrepresentative outcomes are rare on Erdős-Rényi random graphs. The plots show the proportion of minority victories on random graphs drawn from the family $G(100,p)$ as a function of the edge probability $p$ ($x$ axis) for three different scenarios: (a) a minority that is 20% of the entire electorate, (b) 30% minority, and (c) 40% minority. Each data point in a plot is based on ${10}^6$ numerical experiments. For all three scenarios, the peak probability of a minority victory occurs for an intermediate $p$. But note that the minority never wins more than half of the time; the curves lie below the dashed red line at all values of the edge probability $p$.

Figure 3

Introducing community structure to random graphs allows for a prevalence of unrepresentative outcomes within some parameter regions (the diagonal yellow regions). The proportion of unrepresentative outcomes is shown in color as a function of $p_{\text{in}}$ and $p_{\text{out}}$ on stochastic block networks of $N=100$ nodes with minority blocks of sizes (a) $N_{-}=20$, (b) 30, and (c) 40 nodes, for ${10}^6$ simulations.

Figure 4

The proportion of unrepresentative outcomes on stochastic block networks for the special case $p_{\text{in}}=1.$ The probability of a minority victory is plotted as a function of $p_{\text{out}}$, for networks of size $N=10$. Results for three values of $N_{-}$ are shown, corresponding to minority fractions of $20\%$ (red), $30\%$ (orange), and $40\%$ (yellow). The dotted black lines show the analytical expression in Eq. (5), which agrees with numerical results from ${10}^6$ simulations (solid colored lines).

Figure 5

The solid curves show the proportion of unrepresentative outcomes as described by Eq. (5) for constant $\alpha =2/3$ and $N=5,10,20,100$. The stars indicate the location of the maximum on each curve. The probability $p_{\mathrm{out}}$ that maximizes the proportion of unrepresentative outcomes approaches $\alpha$ as $N$ increases. The limiting location of the peak, $p_{\mathrm{out}}=\alpha$, is marked by the gray vertical line.

Figure 6

Examples of networks with heavy-tailed degree distributions with homophily factors (a) $h=0$, (b) $h=0.3$, (c) $h=0.8$, and (d) $h=1$. All networks are of size $N=15$ with minority size $N_{-}=5$. We use the power law exponent $\lambda =2.5$ to generate the degree distribution.

Figure 7

Proportion of minority winning on networks with heavy-tailed degree distributions and size $N={10}^4$, for (a) $N_{-}=2000$, (b) 3000, and (c) 4000 nodes as a function of the homophily factor $h$, for ${10}^3$ simulations.

Figure 8

Example of majority (orange) and minority (purple) node distributions for geometric random networks with radius (a) $r=0.3$, (b) $r=0.5$, and (c) $r=0.8$.

Figure 9

Proportion of minority victories on a geographic random network as a function of the radius of connection $r$ and probability of nodes lying within their preferred half of the square. (a) $h=0.5$ (no homophily), (b) $h=0.75$ (moderate homophily), and (c) $h=1$ (extreme homophily). The results are for networks of size $N=100$, with minority size $N_{-}=30$, for ${10}^4$ simulations.