Male sparrows with large black chest badges are more dominant in the sparrow colony. They guard their mates during the fertile period - around the period when egg-laying is taking place - and copulate with her frequently. But males with large, in addition, join communal courtships more often than males with small bibs, so these dominant males may achieve paternity both though their mate's offspring, and through extra-pair copulations.
Today, in a local farm, I managed to get these shots of communal courtship in House Sparrows. Two or three males were harassing a female, and she managed to peck one of the males (below).
Moller, A. (1990). Sexual behavior is related to badge size in the house sparrow Passer domesticus Behavioral Ecology and Sociobiology, 27 (1) DOI: 10.1007/BF00183309