Any Saturn games with direct 1v1 competition.

[the7k]

Doom (not sure if this version has vs mode)

Interesting - according to this: http://doom.wikia.com/wiki/Sega_Saturn

The US version doesn't have deathmatch multiplayer, but the Japanese and European versions do. Odd.

[pepharytheworm]

Some more non-fighting game options Death Tank Zwei (unlocked in Duke Nukem 3D) and Vatlva.

[fastbilly1]

Doom (not sure if this version has vs mode)

It does not. No US Saturn FPS have deathmatch.

Baku Baku is the only game I would add. It is a fast falling block puzzle game with best 2 out of 3 and attacks based on combos. Also, you can get it for $15-20.

[8bit]

Bust a Move is a fun puzzle game with a vs/competition mode.

Same goes for Baku Baku. Another good puzzle game with 2 player vs mode.

[BogusMeatFactory]

Get Duke Nukem for the Saturn and play Death Tank. Night made.