루지..

I think that, though he actually loves fighting him, Majima also asks Kiryu to win a fight with him (as a prerequisite for giving him favors) to cope with his inability to say a final no to him. It's a way to rationalize his surrender, to use Kiryu's victory as pretext to self-sacrifice for him even though he wouldn't always deserve that level of generosity (because he's either asking too much, doing so opportunistically, showing little consideration for Majima's feelings and circumstances etc.). We can't forget that, when he was very young, Majima had a problem with being sometimes too nice, ending up exploited and manipulated, which - together with other traumas like being kept from reaching Saejima with the carving an eye, Shimano's torture and use of him as a pawn etc. - caused a deep wound and a reactive intensification of his wild and unpredictable traits. His Mad Dog side is not false, not just a meaningless mask, however a softer part of him always remained, which shows especially in his active love for Kiryu no matter the cost, his willingness to give his life for Haruka, his bond with Saejima, his loyalty to Daigo even after he'd sold him out. For this reason, I believe those fighting challenges he gives Kiryu are also a coping strategy not to feel too spineless when he keeps helping him, to justify it to himself with the fact he only let the strongest fighter have his way, respecting a measurable condition he set himself. It gives him a semblance of control over his weakness for Kiryu. Just my interpretation.