We have previously found that a party can waive a waiver argument by failing to raise it. See, e.g., Qiu Ping Li v. Holder, 612 F.3d 603, 604 (7th Cir.2010); United States v. Moore, 563 F.3d 583, 585 (7th Cir.2009). Nevertheless, we have also recognized that the waiver doctrine is “designed for our own protection as much as that of an opposing party, and therefore need not be asserted by a party for us to invoke it.”