Mathematics of Voting and Elections

It is easy to assume that elections are a simple matter of counting votes to find the most popular choice. However, mathematics reveals a far more complex and surprising landscape where the method of counting can be as decisive as the voters’ preferences themselves. From deciding a club president to shaping a national government, mathematical analysis helps us understand whether electoral systems fairly translate voter intent into outcomes and uphold core democratic principles.

Voting Systems: More Than Just a Plurality

The most common system is simple plurality or first-past-the-post, where the candidate with the most votes wins, even without a majority. While simple, it can lead to a winner disliked by a majority of voters. Consider an election with candidates Alice (40%), Bob (35%), and Charlie (25%). Alice wins with 40%, but 60% preferred someone else.

To address this, many systems require a majority. Instant-runoff voting (IRV) or ranked-choice voting (RCV) asks voters to rank candidates. If no one has over 50% of first-choice votes, the candidate with the fewest votes is eliminated, and their votes are redistributed based on those voters' next choices. This process repeats until a candidate achieves a majority. This system aims to produce a more consensus-driven winner but has its own mathematical quirks.

Another major family is score voting or cardinal methods, like approval voting, where you vote for every candidate you find acceptable, or range voting, where you give each candidate a score. The winner is the candidate with the highest total sum. These methods are praised for reducing the incentive to vote strategically against a disliked front-runner.

The Geometry of Gerrymandering and Apportionment

How districts are drawn—a process called districting—profoundly affects representation. Gerrymandering is the deliberate manipulation of district boundaries to favor one party. Mathematically, this exploits geometry and packing and cracking: "packing" opponents' voters into a few districts they win overwhelmingly, and "cracking" their remaining voters across many districts to dilute their influence.

Mathematicians use metrics like the efficiency gap to measure partisan asymmetry in districting plans. A significant, sustained efficiency gap can be evidence of a mathematical tilt. Solutions often involve algorithmic approaches to drawing compact, population-balanced districts that respect communities, though defining "fair" mathematically remains an active challenge.

Apportionment is the related problem of dividing a fixed number of legislative seats (like U.S. House seats) among states or parties based on population or vote share. No method is perfect. The Hamilton/Vinton method (largest remainder) can suffer from the Alabama paradox, where a state can lose a seat when the total number of seats increases. The Huntington-Hill method, currently used for the U.S. House, minimizes relative differences in district size but can favor smaller states. These paradoxes are not flaws in arithmetic but inherent conflicts between fairness criteria.

Strategic Voting and the Arrow’s Theorem Impossibility

Strategic voting or tactical voting occurs when a voter misrepresents their true preference to achieve a better outcome. For example, in a plurality election, a voter who prefers a weak third-party candidate but fears "wasting" their vote might instead vote for a stronger major candidate they dislike less. Not all systems are equally vulnerable to strategy; some, like approval voting, are more strategy-resistant.

The profound mathematical result in this field is Arrow’s impossibility theorem. Economist Kenneth Arrow proved that no ranked-choice voting system (with three or more candidates) can simultaneously satisfy all of these seemingly reasonable fairness criteria:

1. Unanimity (Pareto Efficiency): If every voter prefers A to B, then the group prefers A to B.
2. Non-dictatorship: No single voter can always determine the winner.
3. Independence of Irrelevant Alternatives (IIA): The relative ranking of A vs. B should not change if a third candidate C enters or drops out.

Arrow’s theorem is not a condemnation of democracy but a crucial insight: every voting system is a compromise. Choosing a system means deciding which mathematical properties—and thus which forms of fairness—you prioritize.

Evaluating Fairness with Mathematical Criteria

How do we evaluate which system is "best"? Mathematicians use formal criteria to compare systems:

• Condorcet criterion: If a candidate would beat every other candidate in a head-to-head matchup, that candidate (the Condorcet winner) should win the election. Many systems, including IRV and plurality, can fail this.
• Monotonicity criterion: Receiving more support should never hurt a candidate. Surprisingly, IRV can fail this—it is possible for a candidate to lose after gaining additional first-choice rankings.
• Consistency criterion: If the electorate is divided into two groups and the same candidate wins in each group separately, that candidate should win the combined electorate.

No system satisfies all desirable criteria. Analysis involves weighing trade-offs. For instance, while the Borda count (where points are assigned based on rank) is sensitive to strategic voting, it often does a good job of reflecting the overall strength of preferences across the electorate.

Common Pitfalls

1. Assuming "More Democratic" Means Simpler: Believing that the simplest system (like plurality) is the most fair is a common error. Mathematical analysis shows simplicity often comes at the cost of accurate representation and can encourage strategic voting.
2. Misunderstanding Arrow’s Theorem: The pitfall is thinking the theorem says all voting systems are equally flawed or arbitrary. Its true value is providing a framework to analyze the specific trade-offs inherent in any chosen system.
3. Conflating Apportionment with Districting: Treating them as the same process is incorrect. Apportionment is the mathematical division of seats among units (states), while districting is the geographical drawing of boundaries within a unit. Both are ripe for mathematical manipulation but involve different tools and paradoxes.
4. Ignoring the Role of Geometry in Gerrymandering: Viewing gerrymandering only as a political act, not a mathematical one, overlooks the core mechanism. Effective reform requires understanding metrics like compactness, efficiency gaps, and the power of cracking and packing.

Summary

• The choice of voting system (plurality, ranked-choice, approval) is not neutral; it is a mathematical rule that can determine the winner independently of voter preferences.
• Gerrymandering uses geometric strategies like "packing and cracking" to manipulate electoral maps, while apportionment methods face unavoidable mathematical paradoxes when dividing seats.
• Strategic voting is a rational response to the incentives built into a system, and its prevalence varies by method.
• Arrow’s impossibility theorem mathematically proves that no perfect ranked-choice voting system exists; all systems involve trade-offs between desirable fairness criteria.
• Analyzing elections requires evaluating systems against formal criteria like the Condorcet criterion and monotonicity to understand their specific strengths and weaknesses.