Neither defeated each other though, or at least it was never implied from their duels.
See, this is the one thing that kills me about the Shanks vs Mihawk argument. One clearly has a WSS title, but the other is a Yonko and has never shown to have lost to Mihawk or even gone after the WSS title. We dont even know what Shanks goals are as a whole. We know they stopped fighting because Mihawk pitied Shanks for losing his arm, however there are clear notions in the story that Shanks either just never lost his strength (clashes with Whitebeard, Kaido and Akainu) and is being underestimated.
People can't deny this, its straight up how its presented in the story. We take the WSS as a literal translation of Mihawk being the best in world, UNTIL somebody comes and unseats him (aka, Zoro). Which is exactly what I think is happening here with Shanks. Neither won their duels, yet a "tie" doesnt forgo the WSS of the title.
Its just one of those things we can't really, completely argue because Shanks never was specified to have lost to Mihawk, nor even shown playing that game in the first place.