mud-slinging (e.g. in a political campaign); mud-slinging match; smear campaign; mud wrestle
(engaging in a round of) mutual criticism; repeated attacks on each other
rivalry over winning the right to host an event; competition to lure (attract, invite) (institutions, manufacturing facilities, foreign tourists, etc.) to a locality
Battle of Dan-no-ura (final battle of the Genpei War; 1185)
competitive lobbying; rivalry in campaigning (for and against)