1. The setup: voters, candidates, ballots
Strip every election down to its bones and the same three ingredients are there. A set of voters, a set of candidates, and, from each voter, a ballot that expresses some piece of their preference. A voting method is just a rule for turning the pile of ballots into a single winner (or a ranked list).
A ranking of all candidates from most-preferred to least-preferred. For $n$ candidates, a preference ballot is a strict ordering — no ties. Two voters with identical rankings have the same ballot.
Most voting math is done with a preference schedule: a table that groups voters by the ballot they cast. Suppose 17 voters rank candidates $A$, $B$, $C$:
| Voters | 6 | 5 | 4 | 2 |
|---|---|---|---|---|
| 1st choice | $A$ | $C$ | $B$ | $B$ |
| 2nd choice | $B$ | $B$ | $C$ | $A$ |
| 3rd choice | $C$ | $A$ | $A$ | $C$ |
Six voters rank $A > B > C$, five rank $C > B > A$, and so on. Every method we'll meet operates on a schedule like this — the difference is just which information from the schedule the method uses.
There is no single agreed-upon definition of a fair election. Each method privileges a different fairness criterion: most first-place votes, broadest acceptance, beats-everyone-head-to-head, hardest-to-strategize-against. Comparing methods means comparing the criteria they honor — and the ones they trade away.
2. Plurality
The simplest method: each voter names a single favorite, and the candidate with the most first-place votes wins. Most national elections in the US and UK use it. The math is grade-school arithmetic.
On the schedule above, plurality counts only the top row:
$$ A: 6, \quad B: 4 + 2 = 6, \quad C: 5 $$$A$ and $B$ tie with 6 each; the runner-up $C$ has 5. (Tie-breaking is itself a separate rule — coin flip, runoff, status quo, etc.)
What plurality ignores: everything past first choice. A candidate ranked second by every single voter is no better off than one ranked last by every voter — both get zero plurality points. This is the source of plurality's most famous failure mode.
A majority winner has > 50% of first-place votes. A plurality winner just has more than anyone else. In a 3-way race a plurality winner can have 35% — meaning 65% of voters preferred someone else. Plurality elects the candidate the largest minority agrees on, which is a very different thing from the candidate the majority agrees on.
3. Plurality with runoff
A patch on plurality: if no candidate gets > 50%, the top two advance to a second round and voters choose between them. Used in French presidential elections, many US primaries, and most "two-round" systems worldwide.
Runoff fixes the most obvious case — a winner with 35% support against two opponents who together carry 65% — by forcing the second-round head-to-head. But it does so at the cost of an extra round (expensive, low turnout) and it can still be derailed: a centrist beloved as everyone's second choice might never make it to round two, because plurality eliminates them in round one.
4. Borda count
Use the whole ranking, not just the top. With $n$ candidates, give each ballot's last-place candidate $0$ points, the next-to-last $1$ point, on up to the top choice getting $n - 1$ points. Sum across all ballots; highest total wins.
$$ \text{Borda score}(X) = \sum_{\text{ballots}} (\text{position points for } X) $$Borda's appeal: a candidate who is acceptable to everyone — high on most ballots even if rarely first — beats a polarizing candidate who is some voters' favorite and other voters' last choice. On the 17-voter schedule, with $n = 3$ candidates (so points are 2, 1, 0):
| Candidate | Calculation | Total |
|---|---|---|
| $A$ | $6(2) + 5(0) + 4(0) + 2(1)$ | $14$ |
| $B$ | $6(1) + 5(1) + 4(2) + 2(2)$ | $23$ |
| $C$ | $6(0) + 5(2) + 4(1) + 2(0)$ | $14$ |
$B$ wins comfortably under Borda — even though $B$ tied $A$ in plurality and finished behind $C$ in first-place votes. The reason: $B$ is everyone's second choice, and Borda rewards that.
Borda is vulnerable to strategic voting. If you know your favorite is safe, you can push them up by ranking their strongest rival dead last rather than honestly. Methods that reward "second choices" can also punish you for being honest about them.
5. Condorcet method
A different angle entirely: simulate every possible head-to-head matchup. Candidate $X$ beats candidate $Y$ if a majority of voters rank $X$ above $Y$ on their ballot. The Condorcet winner is a candidate who wins every such head-to-head.
On the 17-voter schedule, the pairwise contests are:
| Matchup | Voters preferring each | Winner |
|---|---|---|
| $A$ vs $B$ | $A$: 6 · $B$: 5 + 4 + 2 = 11 | $B$ |
| $A$ vs $C$ | $A$: 6 + 2 = 8 · $C$: 5 + 4 = 9 | $C$ |
| $B$ vs $C$ | $B$: 6 + 4 + 2 = 12 · $C$: 5 | $B$ |
$B$ beats both $A$ and $C$ head-to-head — $B$ is the Condorcet winner. If forced to pick "the candidate a majority would pick over any single rival," there's no ambiguity.
A Condorcet winner doesn't always exist. With voters whose preferences cycle — $A > B > C$, $B > C > A$, $C > A > B$ — every candidate loses some head-to-head matchup. The collective preference becomes intransitive even though every individual's preferences are transitive. The group has no consistent "favorite."
6. Instant-runoff voting (ranked-choice)
A multi-round runoff conducted on the original ballots — no second election needed. Used in Australian House elections, Maine's federal races, Alaska, and a growing list of US municipalities.
The algorithm:
- Count first-place votes. If anyone has a majority, they win.
- Otherwise, eliminate the candidate with the fewest first-place votes.
- Redistribute each eliminated ballot to its next-highest non-eliminated choice.
- Repeat from step 1.
IRV fixes the spoiler problem that plagues plurality — a minor candidate can run without splitting the vote, because their voters' ballots automatically transfer when they're eliminated. But it has its own peculiar failures.
"Monotonic" means: if I move a candidate up on my ballot, it should never hurt them. IRV can violate this. A candidate sometimes wins on the strength of being eliminated late — and ranking them higher on extra ballots can cause them to survive an earlier round they would have skipped, putting them up against a stronger opponent and losing. Counterintuitive but real.
7. Approval voting
Ditch rankings entirely. Each voter casts a single yes-or-no per candidate — "approve" or "don't" — and the candidate with the most approvals wins.
Approval is mathematically elegant: it's strategy-light (the optimal play is usually to honestly approve everyone you'd be content with), easy to count (it's still just addition), and never has a spoiler effect in the plurality sense — approving an underdog doesn't subtract from your support for a frontrunner. Used in many academic and professional society elections; advocated by formal voting theorists for general adoption.
The drawback: approval treats "barely acceptable" and "absolutely love" identically. A candidate hated by 40% and tolerated by 60% can beat a candidate loved by 50% and hated by 50%, depending on where voters draw their approval threshold.
8. Arrow's impossibility theorem
By 1951, voting theorists had built up a long catalog of methods, each fixing one flaw of an earlier one and introducing two new ones. Kenneth Arrow asked the obvious next question: is there any ranked-ballot method that satisfies a short list of clearly desirable properties simultaneously?
His four properties — informal versions:
- Unanimity (Pareto): if every voter prefers $X$ to $Y$, then the group ranking puts $X$ above $Y$.
- Independence of irrelevant alternatives (IIA): the group's choice between $X$ and $Y$ depends only on how voters rank $X$ vs $Y$ — not on where they place some unrelated $Z$.
- Non-dictatorship: no single voter's preferences determine the outcome regardless of what everyone else thinks.
- Transitivity (universal domain + rationality): the method must produce a coherent group ranking for every possible profile of individual preferences.
For three or more candidates, no ranked-ballot voting method satisfies all four conditions simultaneously. Pick any method; you can point to one of the conditions it violates.
This is not a critique of any particular method — it's a theorem about the space of all possible methods. The lesson isn't "every voting system is bad." The lesson is that "fairness" decomposes into properties that cannot all be honored at once, so designing a voting system means deciding which trade-off to accept. Borda gives up IIA. IRV gives up monotonicity (a related criterion). Plurality fails the majority criterion. The math doesn't let you escape — only choose your loss.
(Approval voting and other cardinal methods sidestep Arrow technically by not using rankings, but Gibbard's and Satterthwaite's theorems extend similar impossibility results to those. The impossibility is structural, not specific.)
9. Common pitfalls
In a 3+ candidate race, the plurality winner often has < 50% support. A 35-33-32 split has a "winner" who is opposed by 65% of voters. Plurality is a most-popular-minority rule, not a majority rule.
It isn't. Real-world IRV elections have had documented non-monotonicity — increased first-place support hurting a candidate. The effect is rare but mathematical, not hypothetical. Don't market IRV as "your ranking will never backfire."
Under Borda or IRV, an irrelevant third candidate joining or leaving the race can flip the winner between the original two. If you ever hear "the spoiler took votes from $X$," that's an IIA violation in action.
No, he didn't. Arrow proved that no aggregation rule can satisfy four specific conditions at once. Real democracies work fine — they just trade off properties that the math says can't coexist. Arrow constrains design choices; it doesn't invalidate them.
10. Worked examples
Example 1 · Plurality, Borda, and Condorcet pick different winners
22 voters rank 5 candidates $A$, $B$, $C$, $D$, $E$:
| Voters | 8 | 6 | 5 | 3 |
|---|---|---|---|---|
| 1st | $A$ | $B$ | $C$ | $D$ |
| 2nd | $D$ | $D$ | $D$ | $C$ |
| 3rd | $B$ | $A$ | $B$ | $B$ |
| 4th | $C$ | $C$ | $A$ | $A$ |
| 5th | $E$ | $E$ | $E$ | $E$ |
Plurality: $A$ wins with 8 first-place votes.
Borda (4-3-2-1-0):
$$ B: 8(2) + 6(4) + 5(2) + 3(2) = 56 $$ $$ D: 8(3) + 6(3) + 5(3) + 3(4) = 69 $$ $$ A: 8(4) + 6(3) + 5(1) + 3(1) = 58 $$$D$ wins Borda — the consensus second choice.
Condorcet: $D$ vs $A$ → 14 voters (those starting $B$, $C$, or $D$) prefer $D$; $D$ wins. $D$ vs $B$ → 16 voters prefer $D$. $D$ vs $C$ → 14 voters prefer $D$. $D$ vs $E$ → all 22. $D$ beats everyone — Condorcet winner.
So $A$, $D$, $D$ depending on method. Same ballots, three different "winners."
Example 2 · The 2000 US Presidential election as plurality failure
In Florida in 2000, Bush received roughly 2,912,790 votes; Gore 2,912,253; Nader 97,488. Bush won the state (and the presidency) by 537 votes under plurality.
Exit polls suggested that if Nader hadn't run, the overwhelming majority of his voters would have picked Gore as second choice. Under any ranking method that uses second-choice information — Borda, IRV, Condorcet — Gore would almost certainly have won Florida.
This is the canonical "spoiler effect": Nader's presence didn't add information about Bush-vs-Gore, but plurality couldn't see past first choice. The result is an IIA violation in the wild — adding an "irrelevant" candidate flipped the head-to-head outcome.
Example 3 · Condorcet paradox
Three voters, three candidates, perfectly cyclic preferences:
| Voter | 1st | 2nd | 3rd |
|---|---|---|---|
| 1 | $A$ | $B$ | $C$ |
| 2 | $B$ | $C$ | $A$ |
| 3 | $C$ | $A$ | $B$ |
Head-to-heads:
- $A$ vs $B$: voters 1 and 3 prefer $A$; voter 2 prefers $B$. $A$ wins.
- $B$ vs $C$: voters 1 and 2 prefer $B$; voter 3 prefers $C$. $B$ wins.
- $C$ vs $A$: voters 2 and 3 prefer $C$; voter 1 prefers $A$. $C$ wins.
So $A > B$, $B > C$, $C > A$. The collective preference is a rock-paper-scissors cycle. No Condorcet winner exists; the group simply has no consistent favorite, even though each individual does. Concrete demonstration that aggregating transitive preferences can produce an intransitive group preference — the seed of Arrow's theorem.
Example 4 · IRV non-monotonicity
17 voters, 3 candidates:
| Voters | 6 | 5 | 4 | 2 |
|---|---|---|---|---|
| 1st | $A$ | $B$ | $C$ | $B$ |
| 2nd | $B$ | $C$ | $A$ | $C$ |
| 3rd | $C$ | $A$ | $B$ | $A$ |
First-place counts: $A: 6$, $B: 7$, $C: 4$. No majority; $C$ has fewest, eliminated. $C$'s 4 ballots transfer to $A$ (their second choice). New counts: $A: 10$, $B: 7$. $A$ wins.
Now suppose 2 of the $B$-first voters change their minds and rank $A$ first instead — moving $A$ up on those ballots. New first-place counts: $A: 8$, $B: 5$, $C: 4$. $C$ is still eliminated and transfers to $A$. $A: 12$, $B: 5$. $A$ still wins — fine. But consider a different scenario where shifting ballots causes a different candidate to be eliminated first, putting $A$ against a stronger opponent. Constructed examples in the literature show this can flip the winner against the candidate who gained support. (See Tideman or the Wikipedia article for a tight numerical example.) The takeaway: ranking a candidate higher can, in pathological cases, defeat them.