The Emperor allowed Mace to defeat him because he knew Mace would want to kill him there and then. With Anakin being sympathetic towards the Emperor he knew that Anakin wouldn't let him and it was this moment that anakin was set up to betray the jedi's and take his place as Darth Vader. Even right before their fight, The Emperor dispatches three Jedi masters with ease.
Still, it seems like he'd have to know they'd send anakin for this to work. For the entire clone-wars saga I can see him playing both sides of the coin because he'd have so much intelligence and either way he'd win on the government political level. However, the jedi are a more personal level, their movements more random, chaotic, and in general less well known. It's hard to predict how many jedi will be sent to deal with something, the skill/power of each jedi, and to be honest it kinda stretches belief for the emperor being on both sides. I mean, I can believe it, its just that with all the media attention around such a political position, all the security staff watching his moves to make sure he stays alive, a large or powerful contingent of jedi trying to find out about him, that he somehow manages to: get the crystals for lightsaber without jedi noticing, master the force to such a high level, acquire and maintain such skill with a lightsaber, and do both hologram and personal meetings with separatist generals/sith. Oh, and train an apprentice or two. All with no one, not even a paparazzi tabloid story finding out. If his skill had deteriorated due to lack of practice, it makes it easier for him to keep his cover. If he kept practicing shouldn't the guards go 'hey that sounds like a lightsaber in there' kind of concern.
Anyway, throwing the fight just seems harder for me to believe than actually lose the fight. To be fair, Mace probably should've just killed him instead of talking or letting him talk and claim it was the heat of battle.