[@Clever Hans] You'd better not be trying to get me to do your math homework.

For simplicity, let's work on one set of dice at a time. Your d8 VS my d8.

We both have the same chance of landing on any particular side. Both of our die have 8 sides, so there are 8 X 8 = 64 different scenarios that all have the exact same chance of occurring.

If I roll an 8, I'll have beaten 7 out of 8 possible outcomes you could roll. 7 beats 6, 6 beats 5, until I roll a 1 which can't beat anything. So we just have to add all my victories together and compare them to the number 64 to see my chances.

7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 28

We both have a 28/64 chance of winning, or 43.75%. The remaining 8 rolls we tied (12.50%) We'll keep this in mind for later.

I don't think it's going to matter too much that a d6 has a slightly higher chance of landing on all of its sides. It has a better chance of landing on smaller numbers and higher numbers. Moreover, 16.6666666...% repeating is hard to do math stuff with compared to the totally chill 12.5% offered by a d8. Regardless, all outcomes should more or less have the same chance of happening, which is what matters.

Same as before, we multiply the sides of our dice together.  

6 X 8 = 48

This time you have a d6. That means that for each side of my die, there's 6 possible outcomes. If I roll a 7 or an 8, I win every time. Then from 6 and below it becomes possible for me to lose. So we can add that up like so:

6 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 27

And for you it's pretty much the exact opposite:

0 + 0 + 0 + 1 + 2 + 3 + 4 + 5 = 15

And then we tie on 6 rolls, which should account for our 48 rolls.

27/48 = 56.25% = my chance of winning

15/48 = 31.25% = your chance of winning

6/48 = 12.5% = the chance we tie

Now

We now know each dice's chances of beating another dice. But it gets a little interesting here because we aren't rolling these dice as sets. We're rolling both our dice at the same time and seeing who has the highest number. As an example, my first d8 might have been a 5, which was higher than the 4 you rolled, but you rolled a 6 on your d6 and my other die was a 1. So you would have won that. But at any given time, I'm only going to have one die that has my highest number, or two dice with an equal number. Ultimately, any of my dice has to compete with any of your dice. so the outcomes are:

My first d8 against your d8
My second d8 against your d8
My first d8 against your d6
My second d8 against your d6

Because there are so many d8s, this wasn't even really worth considering, since 50% of the contests are going to be like one of the two scenarios mentioned above. So that makes this part pretty simple. We're going to divide our %s in half and add them together. And in the end...

I have a (43.75% + 56.25%) ÷ 2 =  50.00% chance to win.

You have a (43.75% + 31.25%) ÷ 2 = 37.50% chance to win.

And we have a 12.5% chance to tie.