A softball or baseball bat with modified restitution characteristic is provided. The bat can comprise a substantially rigid core coupled with a single or multi-piece sleeve. The core can comprise a tip end, a barrel portion, a handle taper, and a handle. A sleeve assembly can be disposed over the barrel portion, and the sleeve assembly can comprise one or more materials capable of impact absorption. The sleeve assembly can include a ring portion positioned about a circumference of the barrel portion, an outer sleeve, and an inner sleeve disposed between the outer sleeve and the barrel portion.