Is there any two-player game for which it is known that a particular player (not just one of the two players) has a winning strategy but no such strategy is known explicitly?
I see that it might be possible to prove a winning strategy by combining a no-draw game with a strategy-stealing argument, but I cannot find an explicit example where the winning strategy is not immediately obvious.
Bonus points for a real-world (rather than just theoretical) game that has this property.
Summing up the remarks above (and using Gerry's link): A classical example of a game where it is known the first player has a winning strategy yet, even for small board sizes, no strategy is known is the game Chomp.