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No they don't, because that's not the point of the game at all.
I know a third party have developed the Conflict game which sort of does what you want, but it's not something Paizo themselves are likely to be interested in.

Tristram |

Expanding on what Enlight_Bystand said: The game is very much designed for people to cooperate through a series of events and challenges, not just combat. Drawing in people and possibly giving them a skewed expectation of the game could be bad (or it could enlighten people as to what a true rpg can encompass).
Other issues: There will be a lot of secret notes passed between players and/or the DM/officiant, particularly if there are casters of any quality, readied actions, or invisible PCs. This can slow down gameplay immensely.
Another issue would be party composition: One group shows up with what would be a balanced 4 man party: Fighter/Cavalier/Pally, Rogue/Ranger, Cleric/Oracle/Witch, Wizard/Sourcerer. The other party has all tier 1 and 2 classes, with maybe someone going Magus for some early action economy breakage. Guess who might win this one?
Things like this are difficult to run and balance since both sides involve fluid/fluctuating options based on player choices/variety with no real way for a DM to control or alter scenarios for balance like they could in a standard game.

Tristram |

The only officially sanctioned event Paizo does is Pathfinder Society.
If you have enough friends who play you could try battles, I would recommend smaller groups (2v2?) and lower level (4-7) and make up the arena secretly so no one knows what to expect. I would also enforce no lead time for casting buffs, if you need to buff you must do it after match start.

Epic Meepo RPG Superstar 2009 Top 16, 2012 Top 32 |
If I'm not mistaken, Paizo actually does run a PvP event at PaizoCon each year (Green Blood on Black Rock, in which players choose monsters to play and fight one another). I think that's plenty as far as officially-sponsored PvP goes, but the game certainly allows for that style of play if you want it.