I've actually tried both, and they both have a certain appeal. The method DMP mentions above is absolute. It keeps things simple, and resolution is quicker; in fact, it also works well with ranged attacks beating initiative, regardless of initiative results. Then there is the other way: If the combatant who loses initiative attacks at 2/1 rate or greater, and has reach, potentially you can have that 1e feel of attacks being split over the the round. It could look something like this:
Halberd Wielder (2/1) vs. Dagger Wielder (1/1)
Dagger wielder wins initiative. The two close in on each other, but because of the WC difference, and despite the loss of initiative, you could have halberd attack, dagger attack, halberd attack.
I'm partial to the latter, even though I've used the former more in practice. I'm curious how other folks might prefer it, as this might require clarification in the new printing.