Melt the butter in a large saucepan over medium heat. Add the shredded red cabbage and stir to coat thoroughly.
Pour in the red wine, balsamic vinegar and sugar and stir everything through. Place the lid on and continue cooking for a further 15 minutes until the liquid has reduced by about one half. If the cabbage has softened to the state you like, but the liquid is not quite reduced, take the lid off to help it along. If the liquid is all gone before the cabbage is finished cooking to your liking, add a splash more of both the wine and balsamic vinegar and keep simmering.