Is there a simple game with asymmetric complexity?
Perhaps a fairly natural game is the following:

Player 1 is placed in the middle of a maze and must reach the exit in order to win.

Player 2 is in the same maze and must collect a set of "components" to build  a radio controller that lets him close the exit (and win).

Deciding the next move from a winning position is easy for Player 1: just follow the shortest path from his current position and the exit; but can be very hard for Player 2.
Indeed if the (remaining) components of the radio controller are on the vertices of a grid graph of $n$ nodes (suppose that Player 2 is himself on an adjacent vertex) and the distance between Player 1 and the exit is exactly $n$, then he must find an Hamiltonian path in order to move.

To make the game more "interactive", we can also add some extra actions to Player 2 that can only cause a polynomial slowdown in the calculation of the next move for Player 1; for example allowing him to block a fixed numbers of corridors of the maze.