Pink Baton is a item that first appeared in the Mario and Sonic at the Olympic Games series. It was first seen on Luigi's art for the game, where he is seen ogling over it, suggesting that the Pink Baton has magical powers to keep the user captivated on it, or that Luigi had romantic relations with it.
Batons are used in relay races as well.
Fighters of Lapis
In Fighters of Lapis, a Pink Baton is a item that is used. It keeps anyone near it transfixed to it, only able to attack very weakly. If Luigi grabs it, his strength is increased. If Daisy is also in the match, she may get jealous.