Why Instant-Runoff Voting Is So Resilient to Coalitional Manipulation: Phase Transitions in the Perturbed Culture

Previous studies have shown that Instant-Runoff Voting (IRV) is highly resistant to coalitional manipulation (CM), though the theoretical reasons for this remain unclear. To address this gap, we analyze the susceptibility to CM of three major voting rules-Plurality, Two-Round System, and IRV-within the Perturbed Culture model. Our findings reveal that each rule undergoes a phase transition at a critical value theta\_c of the concentration of preferences: the probability of CM for large electorates converges exponentially fast to 1 below theta\_c and to 0 above theta\_c. We introduce the Super Condorcet Winner (SCW), showing that its presence is a key factor of IRV's resistance to coalitional manipulation, both theoretically and empirically. Notably, we use this notion to prove that for IRV, theta\_c = 0, making it resistant to CM with even minimal preference concentration.