Polyandry is very rare in birds. Only jacanas are simultaneously so. Females hold "super-territories" encompassing several male territories. Males defend territories. A female may copulate up to 4 males on the same day, so they are probably never certain that they fathered the clutch the care for! Polyandry also occurs in Tasmanian Native Hens and tinamous. Spotted Sandpipers (up to 4 males in succession) and dotterals are SERIALLY POLYANDROUS.
Polyandrous FEMALES are often prettier but sex role changes do not necessitate polyandry--Wilson's Phalaropes are monogamous.
Greater Rheas actually a mixture of polygny and successive polyandry. Males compete for small flocks of females. Dominant ones win harem. The male makes a nest and each female lays several eggs in it (average 28 eggs, record 62!). The females then move on to new male! The males incubate and guard precocial young.
No good theory for polyandry has been proposed. Why is it so rare? A salient feature of polyandrous species is that it MUST NOT BE DISADVANTAGEOUS FOR MALES, so young must be independent at hatching.