A two-round election: n candidates split into two groups, each group selects a winner, the two winners face off. The partition into groups determines the outcome — arrange the groups strategically and you can make your preferred candidate win.
The paper asks: what partition maximizes the probability of a target candidate winning? Under a spatial voting model with cyclic preferences and uniformly distributed ideal points, the answer converges to a precise structure as n grows. The target candidate should be placed in a group containing a cluster of exactly n/5 nearby candidates.
One-fifth. Not one-half (which would dilute the competition), not a minimal group (which would make the final round harder). The optimal manipulation places exactly one-fifth of the candidates in a tight cluster around the target, creating a local advantage in the first round while maintaining a favorable matchup in the final.
The probability of the target winning approaches 1 as the electorate grows. With enough voters, the strategic partition guarantees victory regardless of the candidates' quality or the voters' preferences — the geometry of the partition alone determines the outcome. The manipulation is structural, not informational.
This is manipulability in the formal sense: the election mechanism can be exploited by the entity controlling the partition. The one-fifth constant is not a design parameter but an emergent feature of the interaction between spatial preferences and two-round elimination. It falls out of the optimization, not the setup.