Experiment 18 Voting Rules
Who Actually Won?
Count the same preferences four ways and explore what changes when one voter submits a strategic ballot.
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Start with “Work It Out, Step by Step” below. The optional expert section explains its symbols as you go. For more examples, use the plain-language math guide.
1. Predict, Then Vote
Can the Counting Rule Change the Winner?
You have one vote. Three other groups of voters have the preferences shown in the sandbox. A ballot is what a voter submits; a genuine preference is what they actually want. You may submit a different ranking without changing what you want.
Plurality counts only first choices. Instant runoff repeatedly removes the candidate with the fewest first choices and transfers those votes to each voter’s next remaining choice. Approval counts every candidate a voter marks acceptable. Pairwise majority compares each pair of candidates head to head and looks for someone who beats every other candidate.
2. Reveal the Incentives
Same Voters, Different Rules
| Rule | With your genuine ballot | With your submitted ballot |
|---|
Instant Runoff Rounds
Every Head-to-Head Count
Ties are reported explicitly. Plurality and approval keep all tied leaders. Instant runoff stops if elimination is tied; a real election needs a tie rule before voting starts. Pairwise majority selects a candidate only if it strictly beats every rival.
3. Change One Assumption
Keep Preferences Separate From Strategy
The starting election has ten voters: you and three groups of three. When D is not included, every list skips D. The remaining order stays the same. Each group submits its genuine ranking. For approval, a cutoff says how many top choices the group marks acceptable; rankings by themselves do not tell us this.
Your Submitted Ballot
Clear the checkbox to try the ballot below. Your genuine preferences above remain the basis for your personal outcome and the group comparison.
A cutoff larger than the number of included candidates approves everyone. Changing a setting clears the previous result. This model has complete rankings and no abstentions; an other-voter group may have zero members.
Work It Out, Step by Step
- A majority means more than half. With ten votes, half is five, so six or more is a majority. Five is exactly half.
- A candidate can lead the first-choice count with four votes while six voters prefer someone else first. Plurality requires the most first choices, not a majority.
- For a head-to-head count, compare just two candidates on each list. If B is above A, that voter chooses B in this comparison, even if C is their overall favorite.
- A cycle can happen: a majority prefers A to B, another majority prefers B to C, and another prefers C to A. Then no one beats every rival.
In words: ten divided by two is five, and six is greater than five. The extra space separates two calculations; it is not another operation. The group rank score gives three-candidate first, second, and third places 2, 1, and 0 points. It is an illustrative score, not an additional voting rule here.
For experts: formal model and assumptions
What Is Being Compared?
A Condorcet winner is a candidate who beats every other candidate in separate majority comparisons. One need not exist. The pairwise rule here reports that absence; it does not apply a different method to resolve a cycle. In instant runoff, a tied elimination remains unresolved instead of silently choosing a candidate to remove.
- \(N_{AB}\)In words: the number of votes ranking A above B.
- The small AB is a label for this comparison; it is not A multiplied by B. The reverse label BA counts voters ranking B above A.
In words: A wins the head-to-head comparison only when the number preferring A to B is greater than the number preferring B to A. Equality is a tie. The same complete submitted rankings feed plurality, instant runoff, and pairwise counts. Approval additionally needs approval cutoffs. Genuine ranks, rather than strategic ballots, determine the displayed personal and group comparison.
“Strategic voting” means choosing a ballot to influence the result, possibly differently from one’s genuine preferences. This small experiment compares one voter’s changed ballot; it does not claim that everyone’s best strategic response has been found.
For more examples of the notation, use the math reading guide.
4. Transfer the Lesson
A Club Chooses Its Next Trip
Most club members can accept the beach, but a smaller group strongly prefers the mountains. Would a first-choice ballot and an approval ballot necessarily pick the same trip? What information does a simple ranking leave out?
Compare your reasoning
First choices and acceptable options answer different questions. A ranking says which option comes before another, but not how strongly someone feels or where their acceptable choices end. No single example establishes a universally best voting rule.
Concept reference: MIT’s Mathematics of Voting lesson and its voting systems handout.
A Model Is a Place to Start
These small models make the incentives visible. Their results follow from their stated rules; they are not forecasts of how every person or organization behaves. A simulated strategy is a rule, not a personality.
Scenario links save the controls and random seed. To reproduce an interactive run, make the same choices in the same order. Changing a setting restarts the experiment.