They're not easy to find.
At one extreme, you have the zero-sum pure competition games. Chess and Go are classic examples, but the most cutthroat one I've come across is
Caesar and Cleopatra. Divorce in a box, if you ask me.
At the other extreme, you have the "family game" type, which is pretty much random who wins and who loses. The card game
Fluxx is the best example I can think of here. I've played Fluxx while actively trying to lose, and won despite my best efforts. These can be fun, but they're not intellectually stimulating.
The good two-player games are in a sweet spot between those two extremes. The perfect 2-player game has enough randomness that you won't feel bad if you lose, but enough strategy to make you feel like your actions mattered when you win.
Lost Cities,
Odin's Ravens, and
Jaipur are three such games. They are all card games, and what cards you get is significant in all three games. But all 3 games let you manage your luck in different ways, so having a good strategy is essential.
