Lets assume for a second that bigfoot exists, which I am lead to believe he does because he was in a movie when I was a kid, and everything I saw as kid was definitely true. Also all the people that have reported seeing him have been shotgun toting hunters with epic beards, and we all know that people with beards do not lie.
First lets start with looking at bigfoot who is basically what you would get if you mixed a gorilla with a human being. He is assumed to be brutishly strong and fairly intelligent, please note that bigfoot should not be confused with Adam Glass who currently is starting to resemble a half man / half bear hybrid. Being that bigfoot is reported to live predominantly in forests in Northern America it is assumed that he is pretty good at hunting, coupled with the fact that he has never been caught can also indicate that he either has the ability to turn invisible or that he has acrobatic skills on a par with a ninja.
Now polar bears are a particularly confusing species to me, I have to think that any species that lives in the Arctic circle is either fucking retarded or extremely hardcore. For this comparison I will assume that they are indeed the latter and thus coupled with their extreme bodyweight and strength they would be a formidable foe.
The main weakness of a polar obviously would be the temperature due to it living the vast majority of its life in negative temperatures. Thus for this fight to be fair you would have to choose somewhere that is a balance between the two species natural climates. My vote would be for Scotland as its always damned cold there and there are plenty of forests that suit bigfoot quite well.
So we have looked at the physical and mental attributes of these two species and the next important aspect to me is the music they listen to whilst they train for this mighty fight. I have it on good authority that bigfoots favourite song is “The Forest Whispers My Name” by Cradle of Filth and that in general he enjoys black metal and thrash because he is sick of all the shit that hippys play in his forest.
As for polar bears, they are so far from civilization that they generally lack electricity and so don’t get much chance to listen to music. For some strange reason the vast majority of polar bears listen to rap music and almost all of them love the song “Cold As Ice” by M.O.P. Whether that is cool or not is totally in your hands, but I think not.
So in this fight we have two brutishly strong combatants who given the chance to train for this fight would be even stronger than normal. Bigfoot has a slight edge in terms of intelligence, being that he is a master escape artist and polar bears are too thick to catch a lift on a boat to the Bahamas. On the other hand the polar bear probably has the edge in terms of ferocity, in close quarters fighting my money would definitely be with the polar bear.
My personal opinion would be that the winner would totally depend on the rules of the fight. Should they be forced into a cage fight then I think the polar bear would win due to his ferocity, whereas if they were air dropped into a forest I think bigfoot has the cunning and skills to outwit the polar bear and royally fuck him up.