How do you run a short multiplayer Conspiracy draft tournament?
Since multiplayer games tend to run longer than single-player matches, it's hard to do as many rounds as you would for e.g. a standard pod of 8.  The scheme we've come up with at my LGS is to run two rounds with swiss pairing and per-game prize support:


Do the draft itself.  For convenience, I'm going to presume a perfect 8-person pod here.
Break into two 4-person groups (WotC suggests randomly, but even/odd would also work well here)
Each group plays a single four-person game; award prize support to the top two finishers in each group.  (My LGS is going to do 3 packs for first and 1 for second here, but if you want your draft to perfectly utilize a 36-pack box, then you can go 2/1.)
'Swiss' the players for a second round: the top two players from each of the two round-1 games form a new four-person group, and the bottom two players from the round-1 games also form a new four-person group.
Play out the second round of games, with the same prize support as the first: 2 or 3 packs to the first-place finisher in each group, and 1 pack to the second-place finisher.


This way, (a) everyone gets to play a couple of games with their deck (and don't be surprised if folks play again on their own after the draft); (b) everyone but the bottom couple of finishers gets some measure of prize support (which IMHO is important for a political and 'casual' environment like Conspiracy); and (c) players are incentivized to win their games and not just play for top-2.