Suppose politician C goes negative on politician M. Politician’s M’s support declines..where do his supporters go? If there are just two candidates, they either go to politician C or stay at home. But if there are three or more candidates, they might go to politician A, B, or K etc etc. So, to a first order, it is less profitable to go negative the greater the number of candidates.
This resembles the Holmstrom teams model but with unproductive effort.