(strength of victory): [ \textmargin(a, b) = P[a][b] - P[b][a] ] If ( \textmargin(a, b) > 0 ), then ( a ) beats ( b ).
The Tideman algorithm (Ranked Pairs), invented by Nicolaus Tideman in 1987, answers this by saying: Lock in the strongest landslides first, skip any result that would create a cycle. Imagine a tournament. Candidate A beats B by 52% to 48% (a narrow win). Candidate C beats A by 80% to 20% (a landslide). Tideman argues that the landslide should have more weight in determining the winner than the squeaker. tideman algorithm
1. The Core Problem: The Paradox of Voting Most voting systems (Plurality, IRV, Score) suffer from fundamental flaws: they can elect a candidate who would lose in a head-to-head match against every other candidate. This is the Condorcet Loser problem. (strength of victory): [ \textmargin(a, b) = P[a][b]
Its deep insight is that — a landslide should outweigh a squeaker in resolving circular ties, but never override a direct pairwise majority. This balance between strength and directness makes Tideman one of the most intellectually satisfying voting algorithms ever devised. Candidate A beats B by 52% to 48% (a narrow win)
( G ): A directed acyclic graph (DAG) where an edge ( a \to b ) means "the final ranking has ( a ) above ( b )". 4. The Algorithm Step-by-Step Step 0: Input Each voter submits a ranked ballot (no ties allowed in pure Tideman). Step 1: Tally Pairwise Preferences For every ordered pair ( (a, b) ), count how many voters prefer ( a ) to ( b ). Step 2: Compute Victory Margins For each unordered pair ( a, b ), compute ( \textmargin(a, b) ) if positive. Step 3: Sort Victories by Margin (Descending) Create a list of pairs ( (a, b) ) where ( a ) beats ( b ), sorted from largest margin to smallest.
But there is a more insidious problem: (e.g., A > B, B > C, C > A). Here, no single candidate beats all others head-to-head. The question is: How do we break the cycle fairly?